text: Sections(cid:18)of functions in(cid:19)the close-to-convex family 3 (cid:48)(cid:48) zf (z) −1 ∈ D (2) Re 1 + (cid:48) > , z . f (z) 2 The importance of this class is outlined in [12] and it was also remarked that the F class has a special role on certain problems on the class of harmonic univalent D mappings in (see [12] and the references therein). It is worth remarking that F S∗ S∗ K functions in are neither included in nor includes nor . It is well-known F (cid:40) K (cid:40) S ∈ F that and hence, it is obvious from an earlier observation that for f , | | each s (f)(z) is close-to-convex in z < 1/4. An interesting question is to determine n the largest uniform disk with this property (see Conjecture 1 below). We now recall a recent result of Ponnusamy et al. [12]. F | | Theorem D. Every section of a function in the class is convex in the disk z < 1/6. The radius 1/6 cannot be replaced by a greater one. F In the same article the authors [12] observed that all sections functions of are | | − ≥ close-to-convex in the disk z < 1 3 log n for n 5. Consider n − z z2/2 (3) f (z) = − . 0 (1 z)2 ∈ S∗ ∈ K We see that f / , but f . Also, f is extremal for many extremal 0 F 0 0 ∈ F problems for the class . By investigating the second partial sum of f , the 0 authors conjectured the following. ∈ F | | Conjecture 1. Every section s (f) of f is close-to-convex in the disk z < 1/3 n and 1/3 is sharp. In this article we solve this conjecture in the following form. ∈ F (cid:48) Theorem 1. Every section s (f) of f satisﬁes Re (s (f) (z)) > 0 in the disk | | n n | | z < 1/3. In particular every section is close-to-convex in the disk z < 1/3. The radius 1/3 cannot be replaced by a greater one. We remark that this result is much (cid:80)stronger than the original conjecture. The following lemma is useful in the proof of Theorem 1. ∞ ∈ F Lemma E. [12, Lemma 1] If f(z) = z+ a zn , then the following estimates n=2 n hold: | | ≤ n + 1 ≥ (a) a for n 2. Equality holds for f (z) given by (3) or its rotation. n 0 2 (cid:80) 1 ≤ | (cid:48) | ≤ 1 | | (b) f (z) − for z = r < 1. The bounds are sharp. (1 + r)3 (1 r)3 ∞ | | (c) If f(z) = s (z) + σ (z), with σ (z) = a zk, then for z = r < 1 we n n n k=n+1 k have − | (cid:48) | ≤ n(n + 1)rn+2 2n(n + 2)rn+1 + (n + 1)(n + 2)rn σ (z) − . n 2(1 r)3 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

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text: F

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text: is

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text: the

text: disk

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text: F

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text: radius

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text: replaced

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text: We

text: remark

text: that

text: this

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text: the

text: original

text: conjecture.

text: The

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text: proof

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text: F

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text: Sections of functions in the close-to-convex family 7 1.5 4 1.0 2 0.5 1 2 3 2 4 6 0.5 (cid:45) (cid:45)2 1.0 (cid:45) (cid:45)4 1.5 (cid:45) | | | | (a): H( z < 1/3) (b): H( z < 1/2) 6 6 4 4 2 2 (cid:45)2 2 4 6 (cid:45)2 2 4 6 (cid:45)2 (cid:45)2 (cid:45)4 (cid:45)4 (cid:45)6 (cid:45)6 | | | | (c): H( z < 3/4) (d): H( z < 4/5) | | Figure 1. Images of the disks z < r for r = 1/3, 1/2, 3/4, 4/5, − under the function H(z) = 1/(1 z)3 . (cid:2) (cid:3) − −| | ≥ 1 (12) σn(z) 8 × 3n−1 2n2 + 8n + 9 = k(n) (say). (cid:20) (cid:18) (cid:19) (cid:21) (cid:0) (cid:1) Now − (cid:48) 1 1 k (n) = × − log 2n2 + 8n + 9 + 4n + 8 . 8 3n 1 3 ≥ (cid:48) For n 4, k (n) > 0 and hence k(n) is an increasing function of n. Thus for all ≥ ≥ − n 4, we have k(n) k(4) = 73/216. ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Sections

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text: of

text: functions

text: in

text: the

text: close-to-convex

text: family

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text: (a):

text: H(

text: z

text: (b):

text: H(

text: z

text: (c):

text: H(

text: z

text: (d):

text: H(

text: z

text: Figure

text: Images

text: of

text: the

text: disks

text: z

text: r

text: for

text: r

text: under

text: the

text: function

text: H(z)

text: σn(z)

text: k(n)

text: (say).

text: Now

text: k

text: (n)

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text: For

text: n

text: k

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text: k(n)

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text: 8 S. V. Bharanedhar and S. Ponnusamy Finally, from the relations (9), (11) and (12) it follows that (cid:48) 27 − 73 145 ≥ Re (s (z)) > = > 0 for all n 4. n 64 216 1728 The proof is complete. We end the paper with the following conjecture. ∈ F | | Conjecture 2. Every section s (f) of f is starlike in the disk z < 1/3. n References 1. S. V. Bharanedhar, M. Obradovi´c and S. Ponnusamy, Radii problems for sections of close-to- convex functions, Preprint. 2. D. Bshouty and W. Hengartner, Criteria for the extremality of the Koebe mapping, Proc. Amer. Math. Soc. 111(1991), 403–411. 3. P. L. Duren, Univalent Functions, Springer-Verlag, 1983. 4. R. Fournier and H. Silverman, Radii problems for generalized sections of convex functions, Proc. Amer. Math. Soc. 112(1)(1991), 101–107. 5. A. W. Goodman and I. J. Schoenberg, On a theorem of Szeg¨o on univalent convex maps of the unit circle, J. Analyse Math. 44(1984/85), 200–204. 6. L. Iliev, Classical extremal problems for univalent functions, in Complex analysis (Warsaw, 1979), 89–110, Banach Center Publ., 11 PWN, Warsaw. 7. Y. Miki, A note on close-to-convex functions, J. Math. Soc. Japan. 8(1956), 256–268. 8. M. Obradovi´c and S. Ponnusamy, Partial sums and radius problem for certain class of confor- mal mappings, Sib. Math. J. 52(2)(2011), 291-302. 9. M. Obradovi´c and S. Ponnusamy, Injectivity and starlikeness of sections of a class of univa- lent functions, Complex Analysis and Dynamical Systems V (Israel Mathematics Conference Proceedings (IMCP)) Published by Contemporary (AMS) 591(2013), 195–203. 10. M. Obradovi´c and S. Ponnusamy, Starlikeness of sections of univalent functions, Rocky Moun- tain J. Math., To appear. 11. M. Obradovi´c, S. Ponnusamy and K.-J. Wirths, Coeﬃcient characterizations and sections for some univalent functions, Sib. Math. J.54(1) (2013), 679–696. 12. S. Ponnusamy, S. K. Sahoo and H. Yanagihara, Radius of convexity of partial sums of functions in the close-to-convex family, Nonlinear Anal. 95(2014), 219–228. 13. M. S. Robertson, The partial sums of multivalently star-like functions, Ann. of Math. 42(1941), 829–838. 14. St. Ruscheweyh, On the radius of univalence of the partial sums of convex functions, Bull. London Math. Soc. 4(1972), 367–369. 15. St. Ruscheweyh, Extension of G. Szeg¨o’s theorem on the sections of univalent functions, SIAM J. Math. Anal. 19(6)(1988), 1442–1449. 16. St. Ruscheweyh and T. Sheil-Small, Hadamard products of Schlicht functions and the P´olya- Schoenberg conjecture, Comment. Math. Helv. 48(1973), 119–135. 17. T. Sheil-Small, A note on the partial sums of convex schlicht functions, Bull. London Math. Soc. 2(1970), 165–168. 18. H. Silverman, Radii problems for sections of convex functions, Proc. Amer. Math. Soc. 104(4)(1988), 1191–1196. 19. T. J. Suﬀridge, Extreme points in a class of polynomials having univalent sequential limits, Trans. Amer. Math. Soc. 163(1972), 225–237. 20. G. Szeg¨o, Zur Theorie der schlichten Abbildungen, Math. Ann. 100(1)(1928), 188–211. ##LTLine## ##LTLine## ##LTLine##

text: S.

text: V.

text: Bharanedhar

text: and

text: S.

text: Ponnusamy

text: Finally,

text: from

text: the

text: relations

text: and

text: it

text: follows

text: that

text: Re

text: (s

text: (z))

text: for

text: all

text: n

text: n

text: The

text: proof

text: is

text: complete.

text: We

text: end

text: the

text: paper

text: with

text: the

text: following

text: conjecture.

text: F

text: Conjecture

text: Every

text: section

text: s

text: (f)

text: of

text: f

text: is

text: starlike

text: in

text: the

text: disk

text: z

text: n

text: References

label: reference

text: S.

text: V.

text: Bharanedhar,

text: M.

text: Obradovi´c

text: and

text: S.

text: Ponnusamy,

text: Radii

text: problems

text: for

text: sections

text: of

text: close-to-

text: convex

text: functions,

text: Preprint.

text: D.

text: Bshouty

text: and

text: W.

text: Hengartner,

text: Criteria

text: for

text: the

text: extremality

text: of

text: the

text: Koebe

text: mapping,

text: Proc.

text: Amer.

text: Math.

text: Soc.

text: P.

text: L.

text: Duren,

text: Univalent

text: Functions,

text: Springer-Verlag,

text: R.

text: Fournier

text: and

text: H.

text: Silverman,

text: Radii

text: problems

text: for

text: generalized

text: sections

text: of

text: convex

text: functions,

text: Proc.

text: Amer.

text: Math.

text: Soc.

text: A.

text: W.

text: Goodman

text: and

text: I.

text: J.

text: Schoenberg,

text: On

text: a

text: theorem

text: of

text: Szeg¨o

text: on

text: univalent

text: convex

text: maps

text: of

text: the

text: unit

text: circle,

text: J.

text: Analyse

text: Math.

text: L.

text: Iliev,

text: Classical

text: extremal

text: problems

text: for

text: univalent

text: functions,

text: in

text: Complex

text: analysis

text: (Warsaw,

text: Banach

text: Center

text: Publ.,

text: PWN,

text: Warsaw.

text: Y.

text: Miki,

text: A

text: note

text: on

text: close-to-convex

text: functions,

text: J.

text: Math.

text: Soc.

text: Japan.

text: M.

text: Obradovi´c

text: and

text: S.

text: Ponnusamy,

text: Partial

text: sums

text: and

text: radius

text: problem

text: for

text: certain

text: class

text: of

text: confor-

text: mal

text: mappings,

text: Sib.

text: Math.

text: J.

text: M.

text: Obradovi´c

text: and

text: S.

text: Ponnusamy,

text: Injectivity

text: and

text: starlikeness

text: of

text: sections

text: of

text: a

text: class

text: of

text: univa-

text: lent

text: functions,

text: Complex

text: Analysis

text: and

text: Dynamical

text: Systems

text: V

text: (Israel

text: Mathematics

text: Conference

text: Proceedings

text: (IMCP))

text: Published

text: by

text: Contemporary

text: (AMS)

text: M.

text: Obradovi´c

text: and

text: S.

text: Ponnusamy,

text: Starlikeness

text: of

text: sections

text: of

text: univalent

text: functions,

text: Rocky

text: Moun-

text: tain

text: J.

text: Math.,

text: To

text: appear.

text: M.

text: Obradovi´c,

text: S.

text: Ponnusamy

text: and

text: K.-J.

text: Wirths,

text: Coeﬃcient

text: characterizations

text: and

text: sections

text: for

text: some

text: univalent

text: functions,

text: Sib.

text: Math.

text: S.

text: Ponnusamy,

text: S.

text: K.

text: Sahoo

text: and

text: H.

text: Yanagihara,

text: Radius

text: of

text: convexity

text: of

text: partial

text: sums

text: of

text: functions

text: in

text: the

text: close-to-convex

text: family,

text: Nonlinear

text: Anal.

text: M.

text: S.

text: Robertson,

text: The

text: partial

text: sums

text: of

text: multivalently

text: star-like

text: functions,

text: Ann.

text: of

text: Math.

text: St.

text: Ruscheweyh,

text: On

text: the

text: radius

text: of

text: univalence

text: of

text: the

text: partial

text: sums

text: of

text: convex

text: functions,

text: Bull.

text: London

text: Math.

text: Soc.

text: St.

text: Ruscheweyh,

text: Extension

text: of

text: G.

text: Szeg¨o’s

text: theorem

text: on

text: the

text: sections

text: of

text: univalent

text: functions,

text: SIAM

text: J.

text: Math.

text: Anal.

text: St.

text: Ruscheweyh

text: and

text: T.

text: Sheil-Small,

text: Hadamard

text: products

text: of

text: Schlicht

text: functions

text: and

text: the

text: P´olya-

text: Schoenberg

text: conjecture,

text: Comment.

text: Math.

text: Helv.

text: T.

text: Sheil-Small,

text: A

text: note

text: on

text: the

text: partial

text: sums

text: of

text: convex

text: schlicht

text: functions,

text: Bull.

text: London

text: Math.

text: Soc.

text: H.

text: Silverman,

text: Radii

text: problems

text: for

text: sections

text: of

text: convex

text: functions,

text: Proc.

text: Amer.

text: Math.

text: Soc.

text: T.

text: J.

text: Suﬀridge,

text: Extreme

text: points

text: in

text: a

text: class

text: of

text: polynomials

text: having

text: univalent

text: sequential

text: limits,

text: Trans.

text: Amer.

text: Math.

text: Soc.

text: G.

text: Szeg¨o,

text: Zur

text: Theorie

text: der

text: schlichten

text: Abbildungen,

text: Math.

text: Ann.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: ∞ Finally, one can calculate the expansions of (5.8) near the points λ = 0 and λ = . In the leading order we get (cid:48) (cid:48) − − − − − − → [RI,J(λ; φ)]ii,j,j = δi,i(cid:48)δj,j(cid:48)( λq1/2) m(I,J)φ2i Iq 12(I 2i)(J 2j) (1 + O(λ)) at λ 0 (5.19) (cid:0) (cid:1) and (cid:48) (cid:48) − − − − → ∞ [RI,J(λ; φ)]ii,j,j = δi,i(cid:48)δj,j(cid:48)(λq1/2)m(I,J)φ2i Iq 12(I 2i)(J 2j) 1 + O(λ 1) at λ . (5.20) We could calculate next order corrections in (5.20) and compare it with Kirillov and Reshetikhin → ∞ expansion at λ in [17]. We leave this exercise to the reader. 6 Q-operators (cid:91) The theory of the Q-operators related to the aﬃne algebra U (sl(2)) has been developed in q [42, 43]. Two Q-operators Q±(λ) appear as traces of special monodromy matrices over inﬁnite- dimensional representations of the q-oscillator algebra introduced in Section 2. Similar to the usual transfer-matrices these monodromy matrices are derived from the tensor product of the local L- operators. In this section we construct these local L-operators acting in the (I + 1)-dimensional highest weight module V . We also show that this construction can be naturally gener(cid:98)alized to the I ∈ C case of inﬁnite-dimensional Verma module V + with I . ∈ Z I Lets us ﬁrst assume that I . Then we can introduce the transfer matrix T (λ; φ) as- + ∈ C J,I sociated with the inﬁnite-dimensional Verma module, V +, J acting in the quantum space (cid:2) J (cid:3) M⊗ (cid:98) W = V as I (cid:124) (cid:123)(cid:122) (cid:125) i=1 ⊗ ⊗ T (λ; φ) = Tr R (λ; φ) ... R (λ; φ) . (6.1) J,I J,I J,I V + J M times where the trace is deﬁned as in (2.10). We notice that usually the horizontal ﬁeld is introduced via a global twist in the auxiliary space [42, 43]. However, we prefer to use a local ﬁeld and include it into the deﬁnition of the R-matrix. With such a deﬁnition the transfer-matrix still commutes with the shift operator along the periodic chain. We are going to exploit this fact in our construction of factorized Q-operators in the next publication. Due to the conservation law in (5.8) the tran(cid:77)sfer matrix (6.1) has a block-diagonal form (cid:98) (cid:98) IM (l) T (λ; φ) = T (λ; φ), (6.2) J,I J,I (cid:98) l=0 (l) where for each block T (λ; φ) the sum(cid:88)of in- an(cid:88)d out- indices in the quantum space W is ﬁxed to J,I l, i.e. M M (cid:48) i = i = l. (6.3) k k k=1 k=1 Let us call the subspace in the quantum space W with a ﬁxed l as the l-th sector. ∈ C The direct sum expansion (6.2) is also true for I , when the quantum space becomes inﬁnite- M⊗ dimensional, W = V +. In this case the sum in (6.2) runs from zero to inﬁnity, but all blocks I (cid:98) i=1 with a ﬁxed l are still ﬁnite-dimensional. (I) Using asymptotics (5.19-5.20) one can easily calculate asymptotics of the T (λ; φ) in each J block with a ﬁxed(cid:98) l − T(l) (λ; φ)| → = (−λq1/2)−IMφ−JM −q 12IJM+J−l (I + O(λ)) , (6.4) J,I λ 0 1 φ2MqIM 2l l 13 ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Finally,

text: one

text: can

text: calculate

text: the

text: expansions

text: of

text: near

text: the

text: points

text: λ

text: and

text: λ

text: In

text: the

text: leading

text: order

text: we

text: get

text: [RI,J(λ;

text: φ)]ii,j,j

text: Iq

text: O(λ))

text: at

text: λ

text: and

text: [RI,J(λ;

text: φ)]ii,j,j

text: Iq

text: O(λ

text: at

text: λ

text: We

text: could

text: calculate

text: next

text: order

text: corrections

text: in

text: and

text: compare

text: it

text: with

text: Kirillov

text: and

text: Reshetikhin

text: expansion

text: at

text: λ

text: in

text: We

text: leave

text: this

text: exercise

text: to

text: the

text: reader.

text: Q-operators

text: The

text: theory

text: of

text: the

text: Q-operators

text: related

text: to

text: the

text: aﬃne

text: algebra

text: U

text: has

text: been

text: developed

text: in

text: q

text: Two

text: Q-operators

text: Q±(λ)

text: appear

text: as

text: traces

text: of

text: special

text: monodromy

text: matrices

text: over

text: inﬁnite-

text: dimensional

text: representations

text: of

text: the

text: q-oscillator

text: algebra

text: introduced

text: in

text: Section

text: Similar

text: to

text: the

text: usual

text: transfer-matrices

text: these

text: monodromy

text: matrices

text: are

text: derived

text: from

text: the

text: tensor

text: product

text: of

text: the

text: local

text: L-

text: operators.

text: In

text: this

text: section

text: we

text: construct

text: these

text: local

text: L-operators

text: acting

text: in

text: the

text: (I

text: highest

text: weight

text: module

text: V

text: We

text: also

text: show

text: that

text: this

text: construction

text: can

text: be

text: naturally

text: to

text: the

text: I

text: C

text: case

text: of

text: inﬁnite-dimensional

text: Verma

text: module

text: V

text: with

text: I

text: Z

text: I

text: Lets

text: us

text: ﬁrst

text: assume

text: that

text: I

text: Then

text: we

text: can

text: introduce

text: the

text: transfer

text: matrix

text: T

text: (λ;

text: φ)

text: as-

text: C

text: J,I

text: sociated

text: with

text: the

text: inﬁnite-dimensional

text: Verma

text: module,

text: V

text: J

text: acting

text: in

text: the

text: quantum

text: space

text: J

text: M⊗

text: W

text: V

text: as

text: I

text: T

text: (λ;

text: φ)

text: Tr

text: R

text: (λ;

text: φ)

text: R

text: (λ;

text: φ)

text: J,I

text: J,I

text: J,I

text: V

text: J

text: M

text: times

text: where

text: the

text: trace

text: is

text: deﬁned

text: as

text: in

text: We

text: notice

text: that

text: usually

text: the

text: horizontal

text: ﬁeld

text: is

text: introduced

text: via

text: a

text: global

text: twist

text: in

text: the

text: auxiliary

text: space

text: However,

text: we

text: prefer

text: to

text: use

text: a

text: local

text: ﬁeld

text: and

text: include

text: it

text: into

text: the

text: deﬁnition

text: of

text: the

text: R-matrix.

text: With

text: such

text: a

text: deﬁnition

text: the

text: transfer-matrix

text: still

text: commutes

text: with

text: the

text: shift

text: operator

text: along

text: the

text: periodic

text: chain.

text: We

text: are

text: going

text: to

text: exploit

text: this

text: fact

text: in

text: our

text: construction

text: of

text: factorized

text: Q-operators

text: in

text: the

text: next

text: publication.

text: Due

text: to

text: the

text: conservation

text: law

text: in

text: the

text: matrix

text: has

text: a

text: block-diagonal

text: form

text: IM

text: (l)

text: T

text: (λ;

text: φ)

text: T

text: (λ;

text: φ),

text: J,I

text: J,I

text: (l)

text: where

text: for

text: each

text: block

text: T

text: (λ;

text: φ)

text: the

text: in-

text: out-

text: indices

text: in

text: the

text: quantum

text: space

text: W

text: is

text: ﬁxed

text: to

text: J,I

text: l,

text: i.e.

text: M

text: M

text: i

text: i

text: l.

text: k

text: k

text: Let

text: us

text: call

text: the

text: subspace

text: in

text: the

text: quantum

text: space

text: W

text: with

text: a

text: ﬁxed

text: l

text: as

text: the

text: l-th

text: sector.

text: C

text: The

text: direct

text: sum

text: expansion

text: is

text: also

text: true

text: for

text: I

text: when

text: the

text: quantum

text: space

text: becomes

text: inﬁnite-

text: M⊗

text: dimensional,

text: W

text: V

text: In

text: this

text: case

text: the

text: sum

text: in

text: runs

text: from

text: zero

text: to

text: inﬁnity,

text: but

text: all

text: blocks

text: I

text: with

text: a

text: ﬁxed

text: l

text: are

text: still

text: ﬁnite-dimensional.

text: (I)

text: Using

text: asymptotics

text: one

text: can

text: easily

text: calculate

text: asymptotics

text: of

text: the

text: T

text: (λ;

text: φ)

text: in

text: each

text: J

text: block

text: with

text: a

text: l

text: T(l)

text: (λ;

text: φ)|

text: −q

text: (I

text: O(λ))

text: J,I

text: λ

text: l

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: (or the cyclic case qN = 1) and the BLZ construction. As the ﬁrst step, we need to calculate the local Q-operators acting in the tensor product of the q-oscillator algebra and the highest weight representation with the arbitrary weight 2s. It is known [42, 43, 50–52] that one can construct XXZ (or XXX) Q-operators by taking the inﬁnite spin limit in the auxiliary space of the higher spin transfer-matrix. Our new formula (1.1) suits this purpose perfectly. We also notice here that a 3D approach we employ in this paper is useful for constructing universal R-matrices for higher rank algebras [53]. → ∞ Taking the limit I in (1.1) we derive a generalization of the BLZ Q-operators acting in ∈ C the tensor product of the highest weight Verma modules with the arbitrary weight J . The → Z limit J is non-singular and gives the Q-operator in any ﬁnite-dimensional representation. + The case J = 1 reduces to the previously known BLZ Q-operators. The paper is organized as follows. In Section 2 we recall some basic facts about the U (sl(2)) q and the q-oscillator algebras. We also give a deﬁnition of the terminating basic hypergeometric series and their regularized version, which we use in the paper. In Section 3 we deﬁne the 3D R- matrix and discuss its basic properties following [3]. In Section 4 we consider a two-layer projection and derive a formula for the matrix elements of the U (sl(2)) R-matrix R (λ) acting in the q I,J tensor product of two highest weight representations with integer weights I and J. In Section 4 we discuss the properties of this R-matrix and show that for the case I = 1 it reduces to the standard U (sl(2)) L-operator acting in the (J + 1)- dimensional representation space. Then we q transform the formula for the R-matrix from Section 2 to a remarkably simple formula (1.1) which contains only one summation and can be rewritten as a terminating balanced φ series. Using 4 3 this representation we prove two important symmetry relations for the R-matrix R (λ). We I,J show that this construction can be generalized to the case of inﬁnite-dimensional highest weight ∈ C representations with I, J . In Section 5 we introduce two Q-operators Q±(λ) acting in the tensor product of the highest weight modules with the weight I. Based on the factorization property for the transfer-matrix [42, 43] we derive explicit expressions for the matrix elements of the local Q- ∈ C operators for any values of I (including the inﬁnite-dimensional case I ). We also derive the standard T Q-relation and calculate the Wronskian of its two solutions Q±(λ). In Section 6 we list a standard set of functional relations satisﬁed by higher-spin transfer matrices and Q-operators. Finally, in the Conclusion we summarize all results and outline further directions of research. 2 Conventions First, let us recall some simple facts about the U (sl(2)) algebra. It is generated by three q elements E, F and H with deﬁning relations − − − [qH] qHEq H = q2E, qHF q H = q 2F, [E, F ] = (2.1) [q] and the following Casimir element { } { − } C = [q]2F E + qH+1 = [q]2EF + qH 1 , (2.2) where we used the following notations − − { } − [x] = x x 1, x = x + x 1. (2.3) ∈ C The Casimir element is normally parameterized by a complex number J { } C = qJ+1 . (2.4) ∈ C ∈ Z For any J one can introduce an inﬁnite-dimensional Verma module V + with a basis v , j . J j + We deﬁne the inﬁnite-dimensional representation π+ of U (sl(2)) by the following action on the J q module V + J − − [qj] [qJ j] Hv = (J 2j)v , Ev = v − , F v = v . (2.5) j j j j 1 j j+1 [q] [q] 4 ##LTLine## ##LTLine## ##LTLine##

text: (or

text: the

text: cyclic

text: case

text: qN

text: and

text: the

text: BLZ

text: construction.

text: As

text: the

text: ﬁrst

text: step,

text: we

text: need

text: to

text: calculate

text: the

text: local

text: Q-operators

text: acting

text: in

text: the

text: tensor

text: product

text: of

text: the

text: q-oscillator

text: algebra

text: and

text: the

text: highest

text: weight

text: representation

text: with

text: the

text: arbitrary

text: weight

text: It

text: is

text: known

text: that

text: one

text: can

text: construct

text: XXZ

text: (or

text: XXX)

text: Q-operators

text: by

text: taking

text: the

text: inﬁnite

text: spin

text: limit

text: in

text: the

text: auxiliary

text: space

text: of

text: the

text: higher

text: spin

text: transfer-matrix.

text: Our

text: new

text: formula

text: suits

text: this

text: purpose

text: perfectly.

text: We

text: also

text: notice

text: here

text: that

text: a

text: approach

text: we

text: employ

text: in

text: this

text: paper

text: is

text: useful

text: for

text: constructing

text: universal

text: R-matrices

text: for

text: higher

text: rank

text: algebras

text: Taking

text: the

text: limit

text: I

text: in

text: we

text: derive

text: a

text: generalization

text: of

text: the

text: BLZ

text: Q-operators

text: acting

text: in

text: C

text: the

text: tensor

text: product

text: of

text: the

text: highest

text: weight

text: Verma

text: modules

text: with

text: the

text: arbitrary

text: weight

text: J

text: The

text: Z

text: limit

text: J

text: is

text: non-singular

text: and

text: gives

text: the

text: Q-operator

text: in

text: any

text: ﬁnite-dimensional

text: representation.

text: The

text: case

text: J

text: reduces

text: to

text: the

text: previously

text: known

text: BLZ

text: Q-operators.

text: The

text: paper

text: is

text: organized

text: as

text: follows.

text: In

text: Section

text: we

text: recall

text: some

text: basic

text: facts

text: about

text: the

text: U

text: q

text: and

text: the

text: q-oscillator

text: algebras.

text: We

text: also

text: give

text: a

text: deﬁnition

text: of

text: the

text: terminating

text: basic

text: hypergeometric

text: series

text: and

text: their

text: regularized

text: version,

text: which

text: we

text: use

text: in

text: the

text: paper.

text: In

text: Section

text: we

text: deﬁne

text: the

text: R-

text: matrix

text: and

text: discuss

text: its

text: basic

text: properties

text: following

text: In

text: Section

text: we

text: consider

text: a

text: two-layer

text: projection

text: and

text: derive

text: a

text: formula

text: for

text: the

text: matrix

text: elements

text: of

text: the

text: U

text: R-matrix

text: R

text: (λ)

text: acting

text: in

text: the

text: q

text: I,J

text: tensor

text: product

text: of

text: two

text: highest

text: weight

text: representations

text: with

text: integer

text: weights

text: I

text: and

text: J.

text: In

text: Section

text: we

text: discuss

text: the

text: properties

text: of

text: this

text: R-matrix

text: and

text: show

text: that

text: for

text: the

text: case

text: I

text: it

text: reduces

text: to

text: the

text: standard

text: U

text: L-operator

text: acting

text: in

text: the

text: (J

text: dimensional

text: representation

text: space.

text: Then

text: we

text: q

text: transform

text: the

text: formula

text: for

text: the

text: R-matrix

text: from

text: Section

text: to

text: a

text: remarkably

text: simple

text: formula

text: which

text: contains

text: only

text: one

text: summation

text: and

text: can

text: be

text: rewritten

text: as

text: a

text: terminating

text: balanced

text: φ

text: series.

text: Using

text: this

text: representation

text: we

text: prove

text: two

text: important

text: symmetry

text: relations

text: for

text: the

text: R-matrix

text: R

text: (λ).

text: We

text: I,J

text: show

text: that

text: this

text: construction

text: can

text: be

text: generalized

text: to

text: the

text: case

text: of

text: inﬁnite-dimensional

text: highest

text: weight

text: C

text: representations

text: with

text: I,

text: J

text: In

text: Section

text: we

text: introduce

text: two

text: Q-operators

text: Q±(λ)

text: acting

text: in

text: the

text: tensor

text: product

text: of

text: the

text: highest

text: weight

text: modules

text: with

text: the

text: weight

text: I.

text: Based

text: on

text: the

text: factorization

text: property

text: for

text: the

text: transfer-matrix

text: we

text: derive

text: explicit

text: expressions

text: for

text: the

text: matrix

text: elements

text: of

text: the

text: local

text: Q-

text: C

text: operators

text: for

text: any

text: values

text: of

text: I

text: (including

text: the

text: inﬁnite-dimensional

text: case

text: I

text: We

text: also

text: derive

text: the

text: standard

text: T

text: Q-relation

text: and

text: calculate

text: the

text: Wronskian

text: of

text: its

text: two

text: solutions

text: Q±(λ).

text: In

text: Section

text: we

text: list

text: a

text: standard

text: set

text: of

text: functional

text: relations

text: satisﬁed

text: by

text: higher-spin

text: transfer

text: matrices

text: and

text: Q-operators.

text: Finally,

text: in

text: the

text: Conclusion

text: we

text: summarize

text: all

text: results

text: and

text: outline

text: further

text: directions

text: of

text: research.

text: Conventions

text: First,

text: let

text: us

text: recall

text: some

text: simple

text: facts

text: about

text: the

text: U

text: algebra.

text: It

text: is

text: generated

text: by

text: three

text: q

text: elements

text: E,

text: F

text: and

text: H

text: with

text: deﬁning

text: relations

text: [qH]

text: qHEq

text: H

text: qHF

text: q

text: H

text: q

text: [E,

text: F

text: [q]

text: and

text: the

text: following

text: Casimir

text: element

text: C

text: E

text: qH

text: where

text: we

text: used

text: the

text: following

text: notations

text: [x]

text: x

text: x

text: x

text: x

text: x

text: C

text: The

text: Casimir

text: element

text: is

text: normally

text: parameterized

text: by

text: a

text: complex

text: number

text: J

text: C

text: C

text: Z

text: For

text: any

text: J

text: one

text: can

text: introduce

text: an

text: inﬁnite-dimensional

text: Verma

text: module

text: V

text: with

text: a

text: basis

text: v

text: j

text: J

text: j

text: We

text: deﬁne

text: the

text: inﬁnite-dimensional

text: representation

text: π+

text: of

text: U

text: by

text: the

text: following

text: action

text: on

text: the

text: J

text: q

text: module

text: V

text: J

text: [qj]

text: [qJ

text: j]

text: Hv

text: (J

text: Ev

text: v

text: F

text: v

text: v

text: j

text: j

text: j

text: j

text: j

text: [q]

text: [q]

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: For further convenience let us deﬁne new variables λ and φ and choose parameters in (4.4) as w = λ2, φ = φ2, φ = 1, ψ = 1, ψ = λ. (4.13) h v h v (cid:2) (cid:3) (cid:2) (cid:3) With such a choice of ﬁelds we deﬁne a properly normalized R-matrix by the following expression (cid:48) (cid:48) R (cid:48) (cid:48) R (λ) i ,j = σ (λ)qI (λ) i ,j (4.14) I,J i,j I,J I,J i,j where σ (λ) is symmetric in I, J and deﬁned by the following expression I,J − − − − − σI,J(λ) = ( 1)m(I,J)q 12IJ 12m(I,J)λ m(I,J)(λ2q I J; q2)m(I,J)+1, (4.15) where m(i, j) = min(i, j). (cid:2)Finally, su(cid:3)bstituting (3.8) into (4.4) we arrive at the following explicit formula − − (cid:48) − (cid:48) (cid:48)− − R (λ; φ) i(cid:48),j(cid:48) = δ (cid:48) (cid:48) (−1)m(I,J)φ2i−Iλi−i(cid:48)−m(I,J) qi2+(I i)(J j ) i (i j)+2I+12IJ 12m(I,J) I,J i,j i+j,i +j (cid:88) (cid:88) (q2; q2) (q2; q2) − i I i − − (cid:48)− − − − × − − i I i ( 1)k+l q2k(i j) 2l(J I j+i) (λ2q I J; q2)m(I,J)+1 − − − − qk(k+1)+l(l+1) (1 λ2q I J 2k 2l) k=0 l=0 − − (cid:48) − − − − − (cid:48) × (q 2i, q2+2j; q2) (q 2j ; q2) − (q 2(I i), q2(1+J j); q2) (q 2(J j ); q2) − − k i k l I i l (q2; q2) (q2; q2) k l (4.16) with ≤ (cid:48) ≤ ≤ (cid:48) ≤ 0 i, i I, 0 j, j J . (4.17) One can show that with such a normalization the matrix elements of the R-matrix are the polyno- − ≤ mials in λ and λ 1 of the degree m(I, J). We shall justify the choice of normalization (4.15) in the next section. 5 Properties of the higher spin R-matrix In this section we shall analyze the formula (4.16) and derive a remarkably simple formula for the R-matrix R (λ) in the form of a single sum. I,J ∈ C First let us notice the formula (4.16) can be naturally extended to any values J . This corresponds to the case when the second space becomes an inﬁnite-dimensional Verma module V + (cid:48) ∞ J with indices j, j running form 0 to . In this case we will choose ∈ Z ∈ C m(I, J) = I, I , J . (5.1) + As an example consider the case I = 1. The double sum in (4.16) contains only two nontrivial   terms and we obtain    − − (cid:48) − − (cid:48) (cid:48)− −  (cid:48) (cid:48) δj,j(cid:48)φ 1[λq 1+2J j ] δj,j(cid:48)+1φ 1[qJ j ]qj J2 1 [R (λ; φ)]i ,j = , (5.2) (1,J) i,j (cid:48) − (cid:48) − (cid:48) δj+1,j(cid:48)φ [qj ]q J+21 j δj,j(cid:48)φ [λq 1 2J +j ] (cid:48) i+1,i +1 (cid:48) where i, i = 0, 1. C ⊗ As usual we can deﬁne the L-operator acting in the tensor product 2 V + as a two-by-two J matrix with matrix elements coinciding with the matrix elements of R (λ). To make a connection 1,J to the standard XXZ L-operator we introduce a rescaled spectral parameter µ = λq1/2, set φ = 1 − C and apply a simple similarity transformation D = diag(1, λ 1) in 2. Then we obtain   − − − − µqH/2 µ 1q H/2 µ[q] F q H/2 L(µ) = (5.3) − − − − µ 1[q] qH/2E µq H/2 µ 1qH/2 10 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: For

text: further

text: convenience

text: let

text: us

text: deﬁne

text: new

text: variables

text: λ

text: and

text: φ

text: and

text: choose

text: parameters

text: in

text: as

text: w

text: φ

text: φ

text: ψ

text: ψ

text: λ.

text: h

text: v

text: h

text: v

text: With

text: such

text: a

text: choice

text: of

text: ﬁelds

text: we

text: deﬁne

text: a

text: properly

text: normalized

text: R-matrix

text: by

text: the

text: following

text: expression

text: R

text: R

text: (λ)

text: i

text: ,j

text: σ

text: (λ)qI

text: (λ)

text: i

text: ,j

text: I,J

text: i,j

text: I,J

text: I,J

text: i,j

text: where

text: σ

text: (λ)

text: is

text: symmetric

text: in

text: I,

text: J

text: and

text: deﬁned

text: by

text: the

text: following

text: expression

text: I,J

text: σI,J(λ)

text: I

text: J;

text: where

text: m(i,

text: j)

text: min(i,

text: j).

text: into

text: we

text: arrive

text: at

text: the

text: following

text: explicit

text: formula

text: R

text: (λ;

text: φ)

text: δ

text: i)(J

text: j

text: i

text: (i

text: I,J

text: i,j

text: i+j,i

text: +j

text: i

text: I

text: i

text: i

text: I

text: i

text: j)

text: I

text: j+i)

text: I

text: J;

text: I

text: J

text: (q

text: (q

text: (q

text: i),

text: j);

text: (q

text: j

text: k

text: i

text: k

text: l

text: I

text: i

text: l

text: k

text: l

text: with

text: i,

text: i

text: I,

text: j,

text: j

text: J

text: One

text: can

text: show

text: that

text: with

text: such

text: a

text: normalization

text: the

text: matrix

text: elements

text: of

text: the

text: R-matrix

text: are

text: the

text: polyno-

text: mials

text: in

text: λ

text: and

text: λ

text: of

text: the

text: degree

text: m(I,

text: J).

text: We

text: shall

text: justify

text: the

text: choice

text: of

text: normalization

text: in

text: the

text: next

text: section.

text: Properties

text: of

text: the

text: higher

text: spin

text: R-matrix

text: In

text: this

text: section

text: we

text: shall

text: analyze

text: the

text: formula

text: and

text: derive

text: a

text: remarkably

text: simple

text: formula

text: for

text: the

text: R-matrix

text: R

text: (λ)

text: in

text: the

text: form

text: of

text: a

text: single

text: sum.

text: I,J

text: C

text: First

text: let

text: us

text: notice

text: the

text: formula

text: can

text: be

text: naturally

text: extended

text: to

text: any

text: values

text: J

text: This

text: corresponds

text: to

text: the

text: case

text: when

text: the

text: second

text: space

text: becomes

text: an

text: inﬁnite-dimensional

text: Verma

text: module

text: V

text: J

text: with

text: indices

text: j,

text: j

text: running

text: form

text: to

text: In

text: this

text: case

text: we

text: will

text: choose

text: Z

text: C

text: m(I,

text: J)

text: I,

text: I

text: J

text: As

text: an

text: example

text: consider

text: the

text: case

text: I

text: The

text: double

text: sum

text: in

text: contains

text: only

text: two

text: nontrivial

text: terms

text: and

text: we

text: obtain

text: j

text: j

text: ]qj

text: [R

text: (λ;

text: φ)]i

text: ,j

text: i,j

text: [qj

text: ]q

text: j

text: [λq

text: +j

text: where

text: i,

text: i

text: C

text: As

text: usual

text: we

text: can

text: deﬁne

text: the

text: L-operator

text: acting

text: in

text: the

text: tensor

text: product

text: V

text: as

text: a

text: two-by-two

text: J

text: matrix

text: with

text: matrix

text: elements

text: coinciding

text: with

text: the

text: matrix

text: elements

text: of

text: R

text: (λ).

text: To

text: make

text: a

text: connection

text: to

text: the

text: standard

text: XXZ

text: L-operator

text: we

text: introduce

text: a

text: rescaled

text: spectral

text: parameter

text: µ

text: set

text: φ

text: C

text: and

text: apply

text: a

text: simple

text: similarity

text: transformation

text: D

text: λ

text: in

text: Then

text: we

text: obtain

text: µ

text: µ[q]

text: F

text: q

text: L(µ)

text: µ

text: µq

text: µ

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 19 (cid:54) (cid:54) [19] A.Sharov and R.M.Roth, “Bounds and constructions for granular me- Let us deﬁne cost fun(cid:18)ctions (cid:19)fi(w) for all 0 i n as dia coding,” IEEE Int. Symp. on Information Theory, pp.2343–2347, St. Petersburg, Russia, August 2011. fk(w) = w0 k = 0, [20] Auplu.atSr.,hmAareloldevirataoncndodRRine.Mtgr,e”.aRtPoCrtohec,n.“teAIrml,lepMrrtooovnnetdiCcebolonlouf.,nCdILso,amn2md0u1cn3o..n, sCtrouncttrioolnsanfodrCgroamn-- fk(w) = ∑r wk+i kr +− ri ∀1 (cid:54) k (cid:54) n − r, [21] J.M.G.M.Tolhuizen, “The generalized Gilbert-Varshamov bound is im- i=0 ∀ − (cid:54) plied by Tura´n’s theorem,” IEEE Trans. on Information Theory, vol.43, fk(w) = wk n r < k n. pp.1605–1606, September 1997. [22] R.R.Varshamov, “Estimate of the number of signals in error correcting From (7), any feasible w should satisfy codes,” Dokl. Acad. Nauk SSSR, vol.117, pp.739–741, 1957. (cid:62) [23] R.R.Varshamov and G.M.Tenengolts, “Codes which correct single fk(w) 1 k = 0, asymmetric errors (in Russian),” Automatika i Telemekhanika, vol.26, (cid:62) ∀ (cid:54) (cid:54) − np.2, pp.288–292, 1965. English translation in Automation and Remote fk(w) (cid:62) 1 ∀ 1− k n(cid:54) r, Control, vol.26, no.2, pp.286–290, 1965. fk(w) 0 n r < k n; [24] J.Weber, C.De Vroedt, and D.Boekee, “Bounds and constructions for ∗ binary codes of length less than 24 and asymmetric distance less than And, w = w gives us equalities i(cid:18)n a(cid:19)ll of them. We are re- 6,” IEEE Trans. on Information Theory, vol.34, no.5, pp.1321–1331, quired to show w∗ also minimizes f (w), where September 1988. [25] J.Weber, C.De Vroedt, and D.Boekee, “New upper bounds on the size n n ∑ oTfhecoordye,svoclo.r3r3e,ctpinpg.43as4y–m43m7e,tMricayer1r9o8rs7,”. IEEE Trans. on Information f (w) = wi i . i=0 [26] R.I.White, R.M.H.New, and R.F.W.Pease, “Patterned media: viable route to 50Gb/in2 and up for magnetic recording,” IEEE Trans. Mag- In order to prove the optimality, we show f (w) can be netics, vol.33, no.1, pt.2, pp.990–995, Jan. 1997. written as [27] R.Wood, M.Williams, A.Kavcic, and J.Miles, “The feasibility of mag- · · · nIEetEicErTercaonrsdaincgtioants1o0nteMraabgintsetipcesr, svqoul.a4re5,inncoh.2o,nppc.o9n1v7e–n9ti2o3n,a2l 0m0e9d.ia,” f (w) = y0 f0(w) + y1 f1(w) + + yn fn(w), [28] E.Yaakobi, J.Ma, L.Grupp, P.H.Siegel, S.Swanson, and J.K.Wolf, “Er- where y ’s are some non-negative constants. Hence, for any i ror characterization and coding schemes for ﬂash memories,” Proc. transversal weight w we have Workshop on the Application of Communication Theory to Emerging Memory Technologies, Miami, Florida, December 2010. (cid:62) ∗ ∗ · · · ∗ f (w) y0 f0(w ) + y1 f1(w ) + + yn fn(w ) · · · ∗ = y0 + y1 + + yn−r = f (w ). APPENDIX A We ﬁrst show the choice of y is unique. Then the prob- OPTIMAL TRANSVERSAL WEIGHT FOR Z CHANNEL lem reduces to show the non-negativity of y. Note that the cost functions fi(w) are inner products of w with some non- In this section, we go over lemmas and proofs that has been negative vectors mi, i.e. fi(w) =< mi, w >, where used in section . Lemma 24. For all r ∈ N, |Dm| (cid:54) (2r)m−r+1. 11,, iiff ii == jj,=and0 n − r < i (cid:54) n mij = (i+r), if 1 (cid:54) i (cid:54) n − r, and i (cid:54) j (cid:54) i + r Proof: The proof is based on induction on m. Let us j assume |Di| (cid:54) (2r)i−r+1 for all i (cid:54) m − 1. Therefore, 0, otherwise. × Now, if we form the (n + 1) (n + 1) matrix M with el- { } Dm = −((r −r!1)! Dm−1 + (r −r!2)! Dm−2 + · · · + 0r!! Dm−r) esomlvenintsg MmTijy,=thec,prwohbeleremcof=ﬁn(d(inn)g, (yn)w, ·i·ll·b,e(ne)q)u.ivMaleinst aton | | | | | | 0 1 n (cid:54) r! D−m−1 + r! D−m−2 + · · · + r! Dm−r upper triangular matrix with non-zero elements on the main (r 1)! (r 2)! 0! diagonal, and hence M is non-singular and the solution is (cid:54) r!(2r)m−r r!(2r)m−r−1 · · · r!(2r)m−2r+1 unique. Since M is upper triangular, MTy = c gives us the − + − + + following recursions on y ’s: (r 1)! (r 2)! 0! i − − − − r!(2r) 1 r!(2r) 2 · · · r!(2r) r (cid:18) (cid:19) (cid:18) (cid:19) = (2r)m r+1( − + − + + ) (r 1)! (r 2)! 0! y0 = 0, (cid:54) − − − · · · − − (2r)m r+1(2 1 + 2 2 + + 2 r) 1 n − i∑1 r + j (cid:54) (2r)m−r+1. yi = (r+i i)( i j=max{i−r,1} i yj) (19) ∀ (cid:54) (cid:54) − i : 1 i n r, (cid:18) (cid:19) (cid:18) (cid:19) The idea behind the optimality proof is to write the cost and function f (w) as a non-negative linear combination of some ∗ − ofetahseirblceosptofiunnt cwtihoincshdmeninoitmedizbeys tfhie(mw)alaln, dansdhohwenwce wis∗thaelsoa yi = ni − n∑{r− } r +i j yj. minimizes the cost function and is the desired optimal solu- ∀ − j=(cid:54)max i r,1 tion. i : n r < i n. ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: A.Sharov

text: and

text: R.M.Roth,

text: “Bounds

text: and

text: constructions

text: for

text: granular

text: me-

text: Let

text: us

text: deﬁne

text: cost

font: AGSDVG+URWPalladioL-Ital

text: for

text: all

font: CXWCPL+URWPalladioL-Roma

text: i

text: n

text: as

text: dia

text: coding,”

text: IEEE

text: Int.

text: Symp.

text: on

text: Information

text: Theory,

text: St.

text: Petersburg,

text: Russia,

text: August

text: fk(w)

text: k

text: sCtrouncttrioolnsanfodrCgroamn--

text: fk(w)

text: ∑r

text: wk+i

text: kr

text: ri

text: ∀1

text: k

text: n

text: r,

text: J.M.G.M.Tolhuizen,

text: “The

text: generalized

text: Gilbert-Varshamov

text: bound

text: is

text: im-

text: plied

text: by

text: Tura´n’s

text: theorem,”

text: IEEE

text: Trans.

text: on

text: Information

text: Theory,

text: fk(w)

text: wk

text: n

text: r

text: k

text: n.

text: September

text: R.R.Varshamov,

text: “Estimate

text: of

text: the

text: number

text: of

text: signals

text: in

text: error

text: correcting

text: From

text: any

text: feasible

text: w

text: should

text: satisfy

text: codes,”

text: Dokl.

text: Acad.

text: Nauk

text: SSSR,

text: R.R.Varshamov

text: and

text: G.M.Tenengolts,

text: “Codes

text: which

text: correct

text: single

text: fk(w)

text: k

text: asymmetric

text: errors

text: (in

text: Russian),”

text: Automatika

text: i

text: Telemekhanika,

text: English

text: translation

text: in

text: Automation

text: and

text: Remote

text: fk(w)

text: 1−

text: k

text: r,

text: Control,

text: fk(w)

text: n

text: r

text: k

text: n;

text: J.Weber,

text: C.De

text: Vroedt,

text: and

text: D.Boekee,

text: “Bounds

text: and

text: constructions

text: for

text: binary

text: codes

text: of

text: length

text: less

text: than

text: and

text: asymmetric

text: distance

text: less

text: than

text: And,

text: w

text: w

text: gives

text: us

text: equalities

text: of

text: them.

text: We

text: are

text: re-

text: IEEE

text: Trans.

text: on

text: Information

text: Theory,

text: quired

text: to

text: show

text: w∗

text: also

text: minimizes

text: f

text: (w),

text: where

text: September

text: J.Weber,

text: C.De

text: Vroedt,

text: and

text: D.Boekee,

text: “New

text: upper

text: bounds

text: on

text: the

text: size

text: n

text: n

text: IEEE

text: Trans.

text: on

text: Information

text: f

text: (w)

text: wi

text: i

text: R.I.White,

text: R.M.H.New,

text: and

text: R.F.W.Pease,

text: “Patterned

text: media:

text: viable

text: route

text: to

text: and

text: up

text: for

text: magnetic

text: recording,”

text: IEEE

text: Trans.

text: Mag-

text: In

text: order

text: to

text: prove

text: the

text: optimality,

text: we

text: show

text: f

text: (w)

text: can

text: be

text: netics,

text: Jan.

text: written

text: as

text: R.Wood,

text: M.Williams,

text: A.Kavcic,

text: and

text: J.Miles,

text: “The

text: feasibility

text: of

text: mag-

text: f

text: (w)

text: yn

text: fn(w),

text: E.Yaakobi,

text: J.Ma,

text: L.Grupp,

text: P.H.Siegel,

text: S.Swanson,

text: and

text: J.K.Wolf,

text: “Er-

text: where

text: y

text: ’s

text: are

text: some

text: non-negative

text: constants.

text: Hence,

text: for

text: any

text: i

text: ror

text: characterization

text: and

text: coding

text: schemes

text: for

text: ﬂash

text: memories,”

text: Proc.

text: transversal

text: weight

text: w

text: we

text: have

text: Workshop

text: on

text: the

text: Application

text: of

text: Communication

text: Theory

text: to

text: Emerging

text: Memory

text: Technologies,

text: Miami,

text: Florida,

text: December

text: f

text: (w)

text: yn

text: fn(w

text: yn−r

text: f

text: (w

text: APPENDIX

text: A

text: We

text: ﬁrst

text: show

text: the

text: choice

text: of

text: y

text: is

text: unique.

text: Then

text: the

text: prob-

text: OPTIMAL

text: TRANSVERSAL

text: WEIGHT

text: FOR

text: Z

text: CHANNEL

text: lem

text: reduces

text: to

text: show

text: the

text: non-negativity

text: of

text: y.

text: Note

text: that

text: the

text: cost

text: functions

text: fi(w)

text: are

text: inner

text: products

text: of

text: w

text: with

text: some

text: non-

text: In

text: this

text: section,

text: we

text: go

text: over

text: lemmas

text: and

text: proofs

text: that

text: has

text: been

text: negative

text: vectors

text: mi,

text: i.e.

text: fi(w)

text: mi,

text: w

text: where

text: used

text: in

text: section

text: Lemma

text: For

text: all

text: r

text: N,

text: |Dm|

text: 11,,

text: iiff

text: ii

text: n

text: r

text: i

text: n

text: mij

text: (i+r),

text: if

text: i

text: n

text: r,

text: and

text: i

text: j

text: i

text: r

text: Proof:

text: The

text: proof

text: is

text: based

text: on

text: induction

text: on

text: m.

text: Let

text: us

text: j

text: assume

text: |Di|

text: for

text: all

text: i

text: m

text: Therefore,

text: otherwise.

text: Now,

text: if

text: we

text: form

text: the

text: (n

text: (n

text: matrix

text: M

text: with

text: el-

text: Dm

text: −((r

text: (r

text: Dm−r)

text: esomlvenintsg

text: (yn)w,

text: ·i·ll·b,e(ne)q)u.ivMaleinst

text: aton

text: n

text: r!

text: r!

text: r!

text: Dm−r

text: upper

text: triangular

text: matrix

text: with

text: non-zero

text: elements

text: on

text: the

text: main

text: (r

text: (r

text: diagonal,

text: and

text: hence

text: M

text: is

text: non-singular

text: and

text: the

text: solution

text: is

text: unique.

text: Since

text: M

text: is

text: upper

text: triangular,

text: MTy

text: c

text: gives

text: us

text: the

text: following

text: recursions

text: on

text: y

text: ’s:

text: (r

text: (r

text: i

text: r

text: (r

text: (r

text: r)

text: n

text: r

text: j

text: yi

text: (r+i

text: i)(

text: i

text: i

text: yj)

text: i

text: i

text: n

text: r,

text: The

text: idea

text: behind

text: the

text: optimality

text: proof

text: is

text: to

text: write

text: the

text: cost

text: and

text: function

text: f

text: (w)

text: as

text: a

text: non-negative

text: linear

text: combination

text: of

text: some

text: ofetahseirblceosptofiunnt

text: cwtihoincshdmeninoitmedizbeys

text: tfhie(mw)alaln,

text: dansdhohwenwce

text: wis∗thaelsoa

text: yi

text: ni

text: n∑{r−

text: r

text: +i

text: j

text: yj.

text: minimizes

text: the

text: cost

text: function

text: and

text: is

text: the

text: desired

text: optimal

text: solu-

text: i

text: tion.

text: i

text: n

text: r

text: i

text: n.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: ##LTLine##

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text: ##LTLine##


text: ∞ 5 , and in particular, it can be a code with minimum distance 3, which contradicts the average sphere packing value. (cid:50) Example 3 depicts a directed, i.e. not symmetric, graph where the average sphere packing value does not hold. Next we show an example of a symmetric graph that does not satisfy the average sphere packing value either. Example 4. Assume there are n = k2 vertices partitioned into two groups: the ﬁrst one consists of k vertices and the other − group of the remaining n k vertices. Every vertex from the ﬁrst group is connected (symmetrically) to a set of exactly − − n k = k 1 vertices from the second group such that there k − is no overlap between these k sets. The n k vertices in the G second group are all connected to each other. Thus, the aver- Fig. 2. The graph . 2 age radius-one ball size is · − − √ k k + (n k)(n k + 1) − − √1 ∆1 = n = n 2 n + 3 n > n/2. Therefore, the average sphere packing value is less than 2. However, it is possible to construct a single-error correcting code with the k vertices of the ﬁrst group. (cid:50) Examples 3 and 4 prove that the average sphere packing G value does not hold in all cases. In fact, from Example 4, we Fig. 3. The graph . 3 do not only conclude that it does not hold in general, but also that the ratio between this value and a size of a code can be G ∈ Proof: Since is regular then for all x X, arbitrarily small. However, it is still very interesting to ﬁnd τd∗e(tHr(x(G) , r=)) (cid:54)∆r|X|a.nSdincaeccGordisinaglsotosyLmemmemtraic w2,e whaevehtahvaet some minimal conditions such that this bound holds. ∈ ∆r fgoert aτl∗l(xH(GX,,r)d)e(cid:62)girn|(Xx|). =Th∆errefaonrde,aτcc∗o(Hrdi(nGg,tro))L=em|mX|a. 1, we B. RMeomneomtobneircitthyataandveFcrtaocrtwioniaslaTfrraacntsivoenraslatlrsansversal if w (cid:62) The next example∆prroves that the requirement on t∆hre graph 0 and for 1 (cid:54) i (cid:54) n, G to be symmetric is necessary in order to have equality be- ∑ (cid:62) w 1. tween the sphere packing and the generalized sphere packing y ∈ bound. y Br(xi) A ﬁrst example for choosing a fractional transversal is stated G Example 2. In this example the graph = (X , E ) has six in the next lemma. { }2 (cid:54)2 (cid:54)2 vertices, so X = x , x , x , x , x , x . For 2 i 6, there Lemma 4. The vector w given by 2 1 2 3 4 5 6 is an edge from x to x and ﬁnally there is an edge from x i 1 (cid:54) (cid:54) 1 1 to x2; seeGFig. 2. Therefore, b1(xi) = 2 for all 1 i 6, so wi = min ∈ {deg (x)}, t|he|graph 2 is regular and the sphere packing bound becomes x Brin(xi) r X2 = 3. However, the vector w = (1, 0, 0, 0, 0, 0) is a frac- for 1 (cid:54) i (cid:54) n, is a fractional transversal. 2 tional transversal, which is optimal, and thus the generalized G (cid:62) (cid:54) (cid:54) sphere packing bound of 2 equals 1. (cid:50) Pr∈oof: It is easy to v∈erify that w 0. For every 1 i n, if y Br(xi), then xi Brin(y) and thus In the next example, we show a graph that does not obey 1 (cid:62) 1 taonstwheeravtoerathgee sepahrleireer pqauceksitniognvawlueea. sTkheids pinrotvhiedeIsntarondeugcattiiovne wy = minx∈Brin(y){degr(x)} degr(xi). regarding the validity of the average sphere packing value as Therefore, we get a valid bound. ∑ (cid:62) ∑ 1 Example 3. The graph {G3 = (X3, E3) i}n this example has y∈Br(xi) wy y∈Br(xi) degr(xi) = 1. ﬁve vertices, so X = x , x , x , x , x . There is an edge 3 1 2 3 4 5 from the ﬁrst vertex to all other four vertices; see Fig. 3. The G · · A graph is said to satisfy the monotonicity property, or average size of a ball is 1 5+4 1 = 9/5 and thus the aver- G (cid:62) ∈ ∈ age sphere packing value beco5mes 5 = 25/9. However, the is monotone, if for every r 1, x X and y Br(x), C 9/{5 } G (cid:54) minimum distance of the code = x , x , x , x in is deg (y) deg (x). 2 3 4 5 3 r r ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: and

text: in

text: particular,

text: it

text: can

text: be

text: a

text: code

text: with

text: minimum

text: distance

text: which

text: contradicts

text: the

text: average

text: sphere

text: packing

text: value.

text: Example

text: depicts

text: a

text: directed,

text: i.e.

text: not

text: symmetric,

text: graph

text: where

text: the

text: average

text: sphere

text: packing

text: value

text: does

text: not

text: hold.

text: Next

text: we

text: show

text: an

text: example

text: of

text: a

text: symmetric

text: graph

text: that

text: does

text: not

text: satisfy

text: the

text: average

text: sphere

text: packing

text: value

text: either.

text: Example

text: Assume

text: there

text: are

text: n

text: vertices

text: partitioned

text: into

text: two

text: groups:

text: the

text: ﬁrst

text: one

text: consists

text: of

text: k

text: vertices

text: and

text: the

text: other

text: group

text: of

text: the

text: remaining

text: n

text: k

text: vertices.

text: Every

text: vertex

text: from

text: the

text: ﬁrst

text: group

text: is

text: connected

text: (symmetrically)

text: to

text: a

text: set

text: of

text: exactly

text: n

text: k

text: k

text: vertices

text: from

text: the

text: second

text: group

text: such

text: that

text: there

text: k

text: is

text: no

text: overlap

text: between

text: these

text: k

text: sets.

text: The

text: n

text: k

text: vertices

text: in

text: the

text: G

text: second

text: group

text: are

text: all

text: connected

text: to

text: each

text: other.

text: Thus,

text: the

text: aver-

text: Fig.

text: The

text: graph

text: age

text: radius-one

text: ball

text: size

text: is

text: k

text: k

text: (n

text: k)(n

text: k

text: √1

text: ∆1

text: n

text: n

text: n

text: n

text: Therefore,

text: the

text: average

text: sphere

text: packing

text: value

text: is

text: less

text: than

text: However,

text: it

text: is

text: possible

text: to

text: construct

text: a

text: single-error

text: correcting

text: code

text: with

text: the

text: k

text: vertices

text: of

text: the

text: ﬁrst

text: group.

text: Examples

text: and

text: prove

text: that

text: the

text: average

text: sphere

text: packing

text: G

text: value

text: does

text: not

text: hold

text: in

text: all

text: cases.

text: In

text: fact,

text: from

text: Example

text: we

text: Fig.

text: The

text: graph

text: do

text: not

text: only

text: conclude

text: that

text: it

text: does

text: not

text: hold

text: in

text: general,

text: but

text: also

text: that

text: the

text: ratio

text: between

text: this

text: value

text: and

text: a

text: size

text: of

text: a

text: code

text: can

text: be

text: G

text: Proof:

text: Since

text: is

text: regular

text: then

text: for

text: all

text: x

text: X,

text: arbitrarily

text: small.

text: However,

text: it

text: is

text: still

text: very

text: interesting

text: to

text: ﬁnd

text: τd∗e(tHr(x(G)

text: r=))

text: whaevehtahvaet

text: some

text: minimal

text: conditions

text: such

text: that

text: this

text: bound

text: holds.

text: ∆r

text: fgoert

text: we

text: B.

text: if

text: w

text: The

text: next

text: example∆prroves

text: that

text: the

text: requirement

text: on

text: t∆hre

text: graph

text: and

text: for

text: i

text: n,

text: G

text: to

text: be

text: symmetric

text: is

text: necessary

text: in

text: order

text: to

text: have

text: equality

text: be-

text: w

text: tween

text: the

text: sphere

text: packing

text: and

text: the

text: generalized

text: sphere

text: packing

text: y

text: bound.

text: y

text: Br(xi)

text: A

text: ﬁrst

text: example

text: for

text: choosing

text: a

text: fractional

text: transversal

text: is

text: stated

text: G

text: Example

text: In

text: this

text: example

text: the

text: graph

text: (X

text: E

text: has

text: six

text: in

text: the

text: next

text: lemma.

text: vertices,

text: so

text: X

text: x

text: x

text: x

text: x

text: x

text: x

text: For

text: i

text: there

text: Lemma

text: The

text: vector

text: w

text: given

text: by

text: is

text: an

text: edge

text: from

text: x

text: to

text: x

text: and

text: ﬁnally

text: there

text: is

text: an

text: edge

text: from

text: x

text: i

text: to

text: seeGFig.

text: Therefore,

text: for

text: all

text: i

text: so

text: wi

text: min

text: {deg

text: (x)},

text: t|he|graph

text: is

text: regular

text: and

text: the

text: sphere

text: packing

text: bound

text: becomes

text: x

text: Brin(xi)

text: r

text: However,

text: the

text: vector

text: w

text: is

text: a

text: frac-

text: for

text: i

text: n,

text: is

text: a

text: fractional

text: transversal.

text: tional

text: transversal,

text: which

text: is

text: optimal,

text: and

text: thus

text: the

text: generalized

text: G

text: sphere

text: packing

text: bound

text: of

text: equals

text: Pr∈oof:

text: It

text: is

text: easy

text: to

text: v∈erify

text: that

text: w

text: For

text: every

text: i

text: n,

text: if

text: y

text: Br(xi),

text: then

text: xi

text: Brin(y)

text: and

text: thus

text: In

text: the

text: next

text: example,

text: we

text: show

text: a

text: graph

text: that

text: does

text: not

text: obey

text: taonstwheeravtoerathgee

text: sepahrleireer

text: pqauceksitniognvawlueea.

text: sTkheids

text: pinrotvhiedeIsntarondeugcattiiovne

text: wy

text: minx∈Brin(y){degr(x)}

text: degr(xi).

text: regarding

text: the

text: validity

text: of

text: the

text: average

text: sphere

text: packing

text: value

text: as

text: Therefore,

text: we

text: get

text: a

text: valid

text: bound.

text: Example

text: The

text: graph

text: i}n

text: this

text: example

text: has

text: y∈Br(xi)

text: wy

text: y∈Br(xi)

text: degr(xi)

text: ﬁve

text: vertices,

text: so

text: X

text: x

text: x

text: x

text: x

text: x

text: There

text: is

text: an

text: edge

text: from

text: the

text: ﬁrst

text: vertex

text: to

text: all

text: other

text: four

text: vertices;

text: see

text: Fig.

text: The

text: G

text: A

text: graph

text: is

text: said

text: to

text: satisfy

text: the

text: monotonicity

text: property,

text: or

text: average

text: size

text: of

text: a

text: ball

text: is

text: and

text: thus

text: the

text: aver-

text: G

text: age

text: sphere

text: packing

text: value

text: However,

text: the

text: is

text: monotone,

text: if

text: for

text: every

text: r

text: x

text: X

text: and

text: y

text: Br(x),

text: C

text: G

text: minimum

text: distance

text: of

text: the

text: code

text: x

text: x

text: x

text: x

text: in

text: is

text: deg

text: (y)

text: deg

text: (x).

text: r

text: r

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 10 TABLE II TABLE III Z CHANNEL: UPPER BOUNDS COMPARISON FOR r = 2 Z CHANNEL: UPPER BOUNDS COMPARISON FOR r = 3 n MB ASPV GSPB [24] n MB ASPV GSPB [24] 5 7 5 4 2 5 7 4 2 2 6 12 8 6 4 6 11 6 3 2 7 19 13 9 4 7 17 9 5 2 8 31 21 16 7 8 26 13 7 4 9 51 35 27 12 9 40 20 11 4 10 84 59 46 18 10 63 31 18 6 11 140 101 79 32 11 99 50 29 8 12 238 174 138 63 12 156 80 48 12 13 407 303 243 114 13 248 130 81 18 14 703 532 432 218 14 400 214 136 34 15 1224 942 772 398 15 650 357 231 50 16 2151 1680 1388 739 16 1066 601 395 90 17 3806 3013 2510 1279 17 1764 1020 682 168 18 6780 5433 4562 2380 18 2946 1744 1186 320 19 12153 9845 8327 4242 19 4960 3006 2076 616 20 21902 17924 15260 8069 20 8418 5216 3653 1144 21 39672 32768 28068 14374 21 14395 9108 6462 2134 22 72190 60133 51802 26679 22 24786 15993 11486 4116 23 131914 110740 95904 50200 23 42956 28232 20507 7346 24 241977 204600 178065 ? 24 74902 50081 36768 ? 25 445447 379146 331499 ? 25 131345 89240 66176 ? 26 822696 704555 618679 ? 26 231537 159687 119534 ? 27 1524039 1312642 1157328 ? 27 410164 286866 216639 ? 28 2831211 2451465 2169652 ? 28 729924 517216 393863 ? 29 5273303 4588640 4075740 ? 29 1304514 935722 718180 ? 30 9845788 8607148 7670997 ? 30 2340710 1698286 1313176 ? 31 18424950 16176901 14463616 ? 31 4215629 3091572 2407381 ? 32 34553129 30460760 27317244 ? 32 7618868 5643846 4424196 ? V. LIMITED MAGNITUDE CHANNELS We turn in this section to generalize the Z channel for the non-binary case. In this setup, every symbol can have q val- − { − } ues, 0, 1, . . . , q 1 and we denote [q] = 0, 1, . . . , q 1 . We study the limited magnitude model and focus solely on the single error setup which is carried for two cases. Namely, the error can be asymmetric (Fig. 4(a)) or symmetric (Fig. 4(b)). This error-channel is motivated by the feature of the errors in non-binary ﬂash memories. The cells in ﬂash memories are charged with electrons and due to the inaccuracy in cell- programming and electrons leakage, the charge level of a cell can either increase or decrease by limited magnitude. For more details see for example [5], [6], [13], [17], [28]. A. Asymmetric Errors In the asymmetric non-binary channel, the value of every symbol can only decrease, and in this study we only con- sider the case where the value of each symbol can decrease by G (a) (b) one. The (cid:26)corresponding graph is A,q = (XA,q, EA,q), wh(cid:27)ere XA,q = [q]n and Fig. 4. Two cases of the non-binary channel: (a) asymmetric errors, (b) sym- metric errors. EA,q = (x, y) : x, y ∈ [q]n, x (cid:62) y, ∑n xi = ∑n yi + 1 . i=1 i=1 the monotonicity upper bound and the average sphere packing GsaenitdvHeBdnAGes,qgo,1mA(,eqx,)1x(=x∈){[=qy]nw∈, iHt[sq(]xbna)El:+l xo1f(cid:62).raTdyhi,eu∑shino=ynp1eexrigisr(cid:54)daepsh∑cinri=inb1etydhiibs+yc1aths}ee, vLfoaerlmurem=uan11di0esr. Tthhise smetounpo.tonicity upper bound of the graph GA,q iEs ( A,{q, 1) = (XA,q,∈1, A,q},1), where XA,q,1 = [q]n and G qn+1 A,q,1 = BA,q,1(x) : x [q]n . MB( A,q, 1) = (q − 1)(n + 1). According to the above deﬁnitions it is immediate to verify ∈ (cid:54) G that foGr all y BA,q,1(x), wH(y) wH(x) and thus the Proof: Since the graph A,q is monotone, according to graph A,q is monotone. In the next two lemmas we calculate Lemma 5 the following vector w = (wx)x∈[q]n is a fractional ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: TABLE

text: II

text: TABLE

text: III

text: Z

label: caption

text: CHANNEL:

text: UPPER

text: BOUNDS

text: COMPARISON

text: FOR

text: r

text: Z

text: CHANNEL:

text: UPPER

text: BOUNDS

text: COMPARISON

text: FOR

text: r

text: n

label: table

text: MB

text: ASPV

text: GSPB

text: n

text: MB

text: ASPV

text: GSPB

text: V.

text: LIMITED

text: MAGNITUDE

text: CHANNELS

text: We

text: turn

text: in

text: this

text: section

text: to

text: generalize

text: the

text: Z

text: channel

text: for

text: the

text: non-binary

text: case.

text: In

text: this

text: setup,

text: every

text: symbol

text: can

text: have

text: q

text: val-

text: ues,

text: q

text: and

text: we

text: denote

text: [q]

text: q

text: We

text: study

text: the

text: limited

text: magnitude

text: model

text: and

text: focus

text: solely

text: on

text: the

text: single

text: error

text: setup

text: which

text: is

text: carried

text: for

text: two

text: cases.

text: Namely,

text: the

text: error

text: can

text: be

text: asymmetric

text: (Fig.

text: or

text: symmetric

text: (Fig.

text: This

text: error-channel

text: is

text: motivated

text: by

text: the

text: feature

text: of

text: the

text: errors

text: in

text: non-binary

text: ﬂash

text: memories.

text: The

text: cells

text: in

text: ﬂash

text: memories

text: are

text: charged

text: with

text: electrons

text: and

text: due

text: to

text: the

text: inaccuracy

text: in

text: cell-

text: programming

text: and

text: electrons

text: leakage,

text: the

text: charge

text: level

text: of

text: a

text: cell

text: can

text: either

text: increase

text: or

text: decrease

text: by

text: limited

text: magnitude.

text: For

text: more

text: details

text: see

text: for

text: example

text: A.

text: Asymmetric

text: Errors

text: In

text: the

text: asymmetric

text: non-binary

text: channel,

text: the

text: value

text: of

text: every

text: symbol

text: can

text: only

text: decrease,

text: and

text: in

text: this

text: study

text: we

text: only

text: con-

text: sider

text: the

text: case

text: where

text: the

text: value

text: of

text: each

text: symbol

text: can

text: decrease

text: by

text: G

text: (a)

text: (b)

text: one.

text: The

text: graph

text: is

text: A,q

text: (XA,q,

text: EA,q),

text: XA,q

text: [q]n

text: and

text: Fig.

text: Two

text: cases

text: of

text: the

text: non-binary

text: channel:

text: (a)

text: asymmetric

text: errors,

text: (b)

text: sym-

text: metric

text: errors.

text: EA,q

text: (x,

text: y)

text: x,

text: y

text: [q]n,

text: x

text: y,

text: ∑n

text: xi

text: ∑n

text: yi

text: the

text: monotonicity

text: upper

text: bound

text: and

text: the

text: average

text: sphere

text: packing

text: iHt[sq(]xbna)El:+l

text: Tthhise

text: smetounpo.tonicity

text: upper

text: bound

text: of

text: the

text: graph

text: GA,q

text: iEs

text: A,{q,

text: where

text: [q]n

text: and

text: G

text: x

text: [q]n

text: MB(

text: A,q,

text: (q

text: According

text: to

text: the

text: above

text: deﬁnitions

text: it

text: is

text: immediate

text: to

text: verify

text: G

text: that

text: foGr

text: all

text: y

text: wH(y)

text: wH(x)

text: and

text: thus

text: the

text: Proof:

text: Since

text: the

text: graph

text: A,q

text: is

text: monotone,

text: according

text: to

text: graph

text: A,q

text: is

text: monotone.

text: In

text: the

text: next

text: two

text: lemmas

text: we

text: calculate

text: Lemma

text: the

text: following

text: vector

text: w

text: (wx)x∈[q]n

text: is

text: a

text: fractional

text: ##LTFigure##

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