Resolvent Positive Linear Operators Exhibit the Reduction Phenomenon
aa r X i v : . [ m a t h . SP ] S e p Resolvent Positive Linear OperatorsExhibit the Reduction Phenomenon
Lee Altenberg [email protected]
Abstract
The spectral bound, s ( αA + βV ), of a combination ofa resolvent positive linear operator A and an operatorof multiplication V , was shown by Kato to be convexin β ∈ R . This is shown here to imply, through anelementary lemma, that s ( αA + βV ) is also convex in α >
0, and notably, ∂ s ( αA + βV ) /∂α ≤ s ( A ) when itexists. Diffusions typically have s ( A ) ≤
0, so that fordiffusions with spatially heterogeneous growth or decayrates, greater mixing reduces growth . Models of theevolution of dispersal in particular have found this resultwhen A is a Laplacian or second-order elliptic operator,or a nonlocal diffusion operator, implying selection forreduced dispersal. These cases are shown here to be partof a single, broadly general, ‘reduction’ phenomenon. Keywords : spectral bound — reduction principle— evolution of dispersal — nonlocal dispersal — nonlocaldiffusionThe main result to be shown here is that the growthbound, ω ( mA + V ), of a positive semigroup generated by mA + V changes with positive scalar m at a rate less thanor equal to ω ( A ), where A is also a generator, and V is anoperator of multiplication. Movement of a reactant in aheterogeneous environment is often of this form, where V represents the local growth or decay rate, and m representsthe rate of mixing. Lossless mixing means ω ( A ) = 0, whilelossy mixing means ω ( A ) <
0, so this result implies thatgreater mixing reduces the reactant’s asymptotic growthrate, or increases its asymptotic decay rate. This is afamiliar result when A is a diffusion operator, so whatis new here is the generality shown for this phenomenon.At the root of this result is a theorem by Kingman onthe ‘superconvexity’ of the spectral radius of nonnegativematrices [1]. The logical route progresses from Kingmanthrough Cohen [2] to Kato [3]. The historical route beginsin population genetics.In early theoretical work to understand the evolutionof genetic systems, Feldman and colleagues kept findinga common result from each model they examined [4, 5, Dedicated to Sir John F. C. Kingman on the fiftieth anniversaryhis theorem on the ‘superconvexity’ of the spectral radius [1], whichis at the root of the results presented here.
6, 7, 8, 9, 10, 11] — be they models for the evolution ofrecombination, or of mutation, or of dispersal. Evolutionfavored reduced levels of these processes in populationsnear equilibrium under constant environments, and thisresult was called the
Reduction Principle [10].These results were found for finite-dimensional models.But the same reduction result has also been found in mod-els for the evolution of unconditional dispersal in continu-ous space, in which matrices are replaced by linear oper-ators. This raises the questions of whether this commonresult, discovered in such a diversity of models, reflects asingle mathematical phenomenon. Here, the question isanswered affirmatively.The mathematical underpinnings of the reduction prin-ciple for finite-dimensional models were discovered by SamKarlin [12, 13] (although he did not realize it, and he hadearlier proposed an alternate to the reduction principle —the mean fitness principle [14], which was found to havecounterexamples [15]). Karlin wanted to understand theeffect of population subdivision on the maintenance of ge-netic variation. Genetic variation is preserved if an allelehas a positive growth rate when it is rare, protecting itfrom extinction. The dynamics of a rare allele are approx-imately linear, and of the form x ( t +1) = [(1 − m ) I + m P ] D x ( t ) (1)where x ( t ) is a vector of the rare allele’s frequency amongdifferent population subdivisions, m is the rate of dispersalbetween subdivisions, P is the stochastic matrix represent-ing the pattern of dispersal, and D is a diagonal matrixof the growth rates of the allele in each subdivision. Theallele is protected from extinction if its asymptotic growthrate when rare is greater than 1. This asymptotic growthrate is the spectral radius, ρ ( A ) := max {| λ | : λ ∈ σ ( A ) } , (2)where σ ( A ) is the set of eigenvalues of matrix A .Karlin discovered that for M ( m ) := [(1 − m ) I + m P ],the spectral radius, ρ ( M ( m ) D ), is a decreasing functionof the dispersal rate m , for arbitrary strongly-connecteddispersal pattern: Theorem 1 (Karlin Theorem 5.2, [13, pp. 194–196])
Let P be an arbitrary non-negative irreducible stochastic Lee Altenberg matrix. Consider the family of matrices M ( α ) = (1 − α ) I + α P . Then for any diagonal matrix D with positive terms onthe diagonal, the spectral radius ρ ( α ) = ρ ( M ( α ) D ) is decreasing as α increases (strictly provided D = d I ). Theorem 5.2 means that greater mixing between subdi-visions produces lower ρ ( M ( m ) D ), and if it crosses below1, the allele will go extinct. While this theorem was mo-tivated by the issue of genetic diversity in a subdividedpopulation, the generality of its form applies to any situ-ation where differential growth is combined with mixing. D could just as well represent the investment returns ondifferent assets and P a pattern of portfolio rebalancing.Or D could represent the decay rates of reactant in differ-ent parts of a reactor, and P a pattern of stirring withinthe reactor. In a very general interpretation, Theorem5.2 means that greater mixing reduces growth and hastensdecay .If the dispersal rate m is not an extrinsic parameter,but is a variable which is itself controlled by a gene, then agene which decreases m will have a growth advantage overits competitor alleles. The action of such modifier genesproduces a process that will reduce the rates of dispersalin a population. Therefore, Theorem 5.2 also means that differential growth selects for reduced mixing .In the evolutionary context, the generality of the mixingpattern P in Karlin’s Theorem 5.2 makes it applicable toother kinds of ‘mixing’ besides dispersal. The pattern ma-trix P can just as well refer to the pattern of mutations be-tween genotypes, and then m refers to the mutation rate.Or P can represent the pattern of transmission when twoloci recombine, and then m represents the recombinationrate. The early models for the evolution of recombina-tion and mutation that exhibited the reduction principlein fact had the same form as (1) for the dynamics of a raremodifier allele. Once this was recognized [16, 17, 18], itwas clear that Karlin’s theorem explained the repeated ap-pearance of the reduction result in the different contexts,and generalized the result to a whole class of genetic trans-mission patterns beyond the special cases that had beenanalyzed.The dynamics of movement in space have been longmodeled by infinite-dimensional models, where space iscontinuous and the concentrations of a quantity at eachpoint are represented as a function. The dynamics ofchange in the concentration are modeled as diffusions,where the Laplacian or elliptic differential operator or non-local integral operator takes the place of the matrix P inthe finite-dimensional case. When the substance grows or decays at rates that are a function of its location, the sys-tem is often referred to as a reaction-diffusion. In reaction-diffusion models for the evolution of dispersal, the reduc-tion principle again makes its appearance [19][20, Lemma5.2] [21, Lemma 2.1][22]. In nonlocal diffusion models,again the reduction principle appears [23]. This points tothe possibility of an underlying mathematical unity.Here, a broad characterization of this ‘reduction phe-nomenon’ is established by generalizing Karlin’s theoremto linear operators. The reduction results previously foundfor various linear operators are, therefore, shown to be spe-cial cases of a general phenomenon.This result is actually implicit in Kato’s generalization[3] of Cohen’s theorem [2] on the convexity of the spectralbound of essentially nonnegative matrices with respect tothe diagonal elements of the matrix. It is educed fromKato’s theorem here by means of an elementary ‘dual con-vexity’ lemma.Kato’s goal in [3] was to generalize, from matrices tolinear operators, Cohen’s convexity result [2]: Theorem 2 (Cohen [2])
Let D be diagonal real n × n matrix. Let A be an essentially nonnegative n × n matrix.Then s ( A + D ) is a convex function of D . Here, s ( A + D ) is the spectral bound — the largest realpart of any eigenvalue of A + D . A synonym for the spec-tral bound used in the matrix literature is the spectralabscissa [24, 25, 26]. When the spectral bound is aneigenvalue, it is also referred to as the principal eigenvalue [27], dominant eigenvalue [28], dominant root [29], Perron-Frobenius eigenvalue [30], or
Perron root [31]. ‘Essen-tially nonnegative’ means that the off-diagonal elementsare nonnegative. Synonyms include ‘quasi-positive’ [32],‘Metzler’, ‘Metzler-Leontief’, ‘ML’ [30], and ‘cooperative’[33]:Cohen’s proof relied upon the following theorem ofKingman:
Theorem 3 (Kingman [1])
Let A be an n × n matrixwhose elements, A ij ( θ ) , are non-negative functions of thereal variable θ , such that they are ‘superconvex’, i.e. foreach i, j , either log A ij ( θ ) is convex in θ , or A ij ( θ ) = 0 for all θ .Then the spectral radius of A is also superconvex in θ . Kato generalized Cohen’s result to linear operators byfirst generalizing Kingman’s theorem. Before presentingKato’s theorem, some terminology needs to be introduced: X represents an ordered Banach space or its complexifi-cation. X + represents the proper, closed, positive cone of X , as-sumed to be generating and normal (see [3]). esolvent Positive Linear Operators Exhibit the Reduction Phenomenon B ( X ) represents the set of all bounded linear operators A : X X . A is a positive operator if AX + ⊂ X + . The resolvent of A is R ( ξ, A ) := ( ξ − A ) − , the operatorinverse of ξ − A , ξ ∈ C . The resolvent set ̺ ( A ) ⊂ C are those values of ξ for which ξ − A is invertible. The spectrum of A ∈ B ( X ), σ ( A ), is the complement ofthe resolvent set, ̺ ( A ).The spectral bound of closed linear operator A , not neces-sarily bounded, is s ( A ) := (cid:26) sup { Re( λ ) : λ ∈ σ ( A ) } if σ ( A ) = ∅−∞ if σ ( A ) = ∅ . The type (growth bound) of an infinitesimal generator, A ,of a strongly continuous ( C ) semigroup, { e tA : t > } , is ω ( A ) := lim t →∞ t log k e tA k . Generally, −∞ ≤ s ( A ) ≤ ω ( A ) < + ∞ , but condi-tions for s ( A ) = ω ( A ) or s ( A ) < ω ( A ) are part ofa more involved theory for the asymptotic growth ofsemigroups (see [34]). Definition 1 A is resolvent positive if there is ξ suchthat ( ξ , ∞ ) ⊂ ̺ ( A ) and R ( ξ, A ) is positive for all ξ > ξ [35]. The relationship of the resolvent positive property toother familiar operator properties includes the followinglist of key results:1. If A generates a C -semigroup T t , then T t is positivefor all t ≥ A is resolvent positive [36,p. 188].2. If A is a resolvent positive operator defined densely on X = C ( S ), the Banach space of continuous complex-valued functions on compact space S , then A gener-ates a positive C -semigroup [36, Theorem 3.11.9].3. If A is resolvent positive and its domain, D ( A ) ⊂ X ,is dense in X , then for every f ∈ D ( A ), there existsa unique solution, u ( t ) ∈ D ( A ) for all t ≥ u ∈ C ([0 , ∞ ) , X ), to the Cauchy problem [35, Theorem7.1] ∂u∂t = Au ( t ) ( t ≥ , u (0) = f.
4. If A is resolvent positive then: s ( A ) < + ∞ ; if σ ( A )is nonempty, i.e. −∞ < s ( A ), then s ( A ) ∈ σ ( A ); if ξ ∈ R ∩ ̺ ( A ) yields R ( ξ, A ) ≥ ξ > s ( A ) [3] [36,Proposition 3.11.2]. 5. Differential operators higher than second order arenever resolvent positive [37, Corollary 2.3][38].6. Particular cases of resolvent positive operators in-clude(a) second-order elliptic operators A = n X j,k =1 a jk ( x ) ∂ ∂x j ∂x k + n X j =1 b j ( x ) ∂∂x j + c ( x ) , where the matrix (cid:2) a jk ( x ) (cid:3) nj,k =1 is symmetric andpositive-definite for each x , and appropriate reg-ularity conditions hold for the domain and coef-ficients (e.g. [39],[3],[40]).(b) Linear integral operators A on X = C (Ω) de-fined by( Af )( x ) := Z Ω K ( x, y ) f ( y ) dy + b ( x ) f ( x ) , where K ∈ C (Ω × Ω , R + ), Ω ⊂ R n is bounded,and K ( x, y ) > b ( x ) are measurable functionsfor x, y ∈ Ω [23, 41, 42]. A resolvent positivecombination of integral and differential operatoris analyzed in [43].Kato’s generalization of Cohen’s theorem is as follows.
Theorem 4 (Generalized Cohen’s theorem [3])
Consider X = C ( S ) (continuous functions on a com-pact Hausdorff space S ) or X = L p ( S ) , l ≤ p < + ∞ ,on a measure space S , or more generally, let X be theintersection of two L p -spaces with different p ’s and differ-ent weight functions. Let A : X X be a linear operatorwhich is resolvent positive. Let V be an operator of mul-tiplication on X represented by a real-valued function v ,where v ∈ C ( S ) for X = C ( S ) , or v ∈ L ∞ ( S ) for theother cases.Then s ( A + V ) is a convex function of V . If in particular A is a generator, then both s ( A + V ) and ω ( A + V ) areconvex in V . Results
Theorem 5 (Generalized Karlin’s theorem)
Let A be a resolvent positive linear operator, and V be anoperator of multiplication, under the same assumptions asTheorem 4.Then for m > ,1. s ( m A + V ) is convex in m;2. For each m > , either(a) s (( m + d ) A + V ) < s ( m A + V )+ d s ( A ) ∀ d > ,or Lee Altenberg (b) s (( m + d ) A + V ) = s ( m A + V )+ d s ( A ) ∀ d > ;3. In particular, when s ( A ) = 0 then s ( m A + V ) is non-increasing in m (the ‘reduction phenomenon’), andwhen s ( A ) < then s ( m A + V ) is strictly decreasingin m ;4. Whenever dd m s ( m A + V ) exists, then dd m s ( m A + V ) ≤ s ( A ) . (3) If A is a generator of a C -semigroup, then the above re-lations on s ( m A + V ) also apply to the type ω ( m A + V ) . Proof:
We consider the general form φ ( α, β ) := s ( αA + βV ) or ω ( αA + βV ) (4)where α > , β ∈ R . Kato [3] explicitly shows that φ (1 , β )is convex in β (which he points out is equivalent to vary-ing V ). Lemma 1 (to follow) shows that this implies theproperties asserted above regarding the effects of varying m on s ( m A + V ) = φ ( m, (cid:3) Lemma 1 (Dual Convexity)
Let f : R × R R , bejointly continuous. For x > and y ≥ , let f have thefollowing properties: f ( αx, αy ) = αf ( x, y ) , for α > , (5) and f ( x, y ) is convex in y. (6) Then for x > :1. f ( x, y ) is convex in x , for y > ;2. For each x > , either(a) f ( x + d, < f ( x,
1) + d f (1 , ∀ d > ; or(b) f ( x + d,
1) = f ( x,
1) + d f (1 , ∀ d > ;3. When it exists, ∂∂x f ( x, ≤ f (1 , .If, in the above, f ( x, y ) is strictly convex in y , then f ( x, y ) is strictly convex in x , and f ( x + d, < f ( x, d f (1 , .The results is unchanged if the inequalities on y are re-versed. Proof: f ( x, y ) is convex in x , for y > f ( αx, αy ) = αf ( x, y ) allows a set of rescal-ings that transform convexity in y into convexity in x .It is perhaps worth noting that this relation is actuallya homomorphism, which can be put into a more familiar form by defining product x ⋆ y := f ( x, y ), and function ψ ( x ) := αx , which gives ψ ( x ) ⋆ ψ ( y ) = ψ ( x ⋆ y ).For the following derivations, the constraints are y = 0, y , y = 0 have the same sign as y , and 0 < m < { y, y , y , m, − m ,(1 − m ) y + my } are nonzero and all have the same sign,so that division with them is defined, and their ratios donot change sign when the sign of y is reversed.Convexity of f in y gives(1 − m ) f ( x, y ) + mf ( x, y ) ≥ f ( x, (1 − m ) y + my ) , (7)for m ∈ (0 , y = y . Using (5) with substitutions α = y /y , α = y /y , and α = [(1 − m ) y + my ] /y in theterms in (7), where y ∈ (0 , ∞ ), yields:(1 − m ) y y f (cid:0) xyy , y (cid:1) + m y y f (cid:0) xyy , y (cid:1) ≥ (1 − m ) y + my y f (cid:0) xy (1 − m ) y + my , y (cid:1) . (8)Let x := xy/y and x := xy/y represent the rescaledarguments for f on the left side of (8). We see that x , x ∈ (0 , ∞ ) since x ∈ (0 , ∞ ) and y, y , y = 0 have the samesign.We try the ansatz that x and x can be combined con-vexly to yield the third rescaled argument on the right sideof (8): xy (1 − m ) y + my = (1 − h ) x + hx =(1 − h ) xyy + h xyy . The ansatz has solution h = my (1 − m ) y + my , and 1 − h = (1 − m ) y (1 − m ) y + my . Note that h ∈ (0 ,
1) is assured because y and y have thesame sign, y = y , and m ∈ (0 , φ := [(1 − m ) y + my ] /y . Then φ > y, y , y all have the same sign. Substitution gives(1 − m ) y /y = (1 − h ) φ , and my /y = hφ , and (8) becomes:(1 − h ) φf ( x , y ) + hφf ( x , y ) ≥ φf ((1 − h ) x + hx , y ) . After dividing both sides by φ > − h ) f ( x , y )+ hf ( x , y ) ≥ f ((1 − h ) x + hx , y ) , (9)which is convexity in x . The case of strict convexityfollows by substituting > for ≥ throughout.2. Either f ( x + d, < f ( x,
1) + d f (1 , ∀ d > , or f ( x + d,
1) = f ( x,
1) + d f (1 , ∀ d > . The strategy will be to show first that f ( x + d, ≤ f ( x,
1) + d f (1 , f ( x + d, 0) for any d > 0, then it is true for all d > f ( x + d, 1) from ever returningto the line f ( x, 1) + d f (1 , 0) for d > esolvent Positive Linear Operators Exhibit the Reduction Phenomenon x, d > 0, we have the equivalences f ( x + d, ≤ f ( x, 1) + d f (1 , ⇐⇒ (10)( x + d ) f (cid:0) , x + d (cid:1) ≤ x f (cid:0) , x (cid:1) + d f (1 , ⇐⇒ f (cid:0) , x + d (cid:1) ≤ xx + d f (cid:0) , x (cid:1) + dx + d f (1 , . (11)Since the y arguments for f ( x, y ) in (11) are related byconvex combination,1 x + d = xx + d x + (cid:0) − xx + d (cid:1) ∗ , then (11) is just a statement of the convexity of f ( x, y ) in y , as hypothesized. The case of strict convexity follows bysubstituting < for ≤ , throughout.Now, given x > 0, suppose that for some d > f ( x + d , < f ( x, 1) + d f (1 , . (12)We shall see that convexity then prevents f ( x + d, 1) fromever returning to the line f ( x, 1) + d f (1 , 0) for d > x < x + d < x + d < x + d 1) and F ≡ f (1 , g ( x + d ) ≤ (cid:0) − d d (cid:1) g ( x ) + d d g ( x + d ) < (cid:0) − d d (cid:1) g ( x ) + d d ( g ( x ) + d F ) = g ( x ) + d F. and, by (9), (12), and (10) (line 3 below), g ( x + d ) ≤ d − d d − d g ( x + d ) + d − d d − d g ( x + d ) < d − d d − d ( g ( x ) + d F ) + d − d d − d g ( x + d ) ≤ d − d d − d ( g ( x ) + d F ) + d − d d − d ( g ( x ) + d F ) ⇐⇒ g ( x + d )( d − d ) < ( d − d ) ( g ( x ) + d F ) + ( d − d )( g ( x ) + d F )= ( d − d ) g ( x ) + d ( d − d ) F ⇐⇒ g ( x + d ) < g ( x ) + d F. When it exists, ∂∂x f ( x, ≤ f (1 , . Rearrangement of (10) gives f ( x + d, − f ( x, d ≤ f (1 , . Hence when the limit exists,lim d → f ( x + d, − f ( x, d = ∂f ( x, ∂x ≤ f (1 , . (cid:3) Remark 1 It would be clearly desirable to characterizethe conditions for strict convexity in Kato’s theorem, sothat by Lemma 1, one would obtain strict convexity inTheorem 5, item 1, and strict monotonicity in items 3 and4. Indeed, item 2 is the best that can be offered in theway of strict inequality without strict convexity. But theproblem is more technical and is deferred to elsewhere.It is reasonable, nevertheless, to conjecture that theproperties which produce strict convexity in the matrixcase [44, Theorem 4.1] [45, Theorem 1.1] extend to theirBanach space versions: i.e. for a > 0, when resolvent pos-itive operator A is irreducible [46, p. 250] [47, p. 41], then s ( αA + βV ) is strictly convex in β if and only if V is nota constant scalar. A Third Proof of Karlin’s Theorem 5.2 Karlin’s proof was based on the Donsker-Varadhan vari-ational formula for the spectral radius [48]. Kirkland et al.[49] recently discovered another proof using entirely struc-tural methods. A third distinct proof of Karlin’s theoremis seen here by application of Lemma 1 to Cohen’s the-orem, combined with Friedland’s equality condition [44,Theorem 4.1] (see also [45] for a different proof), as fol-lows.The expression in Karlin’s Theorem 5.2 can be put inthe form used in Theorem 5: M ( m ) D = [(1 − m ) I + m P ] D = m ( P − I ) D + D = α A + β D , where A = ( P − I ) D , α = m , and β = 1.Since e ⊤ ( P − I ) D = ( e ⊤ − e ⊤ ) D = , we see that s ( A ) = s (( P − I ) D ) = 0. Cohen’s theorem gives that s ( α A + β D ) is convex in β , and thus by Lemma 1, s ( α A + β D ) is convex and non-increasing in α . Applica-tion of Lemma 1 therefore yields that ρ ( M ( m ) D ) is convexand non-increasing in m , and d ρ ( M ( m ) D ) / d m ≤ 0, thederivative existing for all m > M ( m ) D is irre-ducible.From Friedland [44, Theorem 4.1], strict convexity in β occurs if P is irreducible and D = c I , for any c > 0. ByLemma 1 this implies s ( M ( m )) is strictly decreasing andstrictly convex in m . (cid:3) Remark 2 The core of Kirkland et al.’s proof is theirLemma 4.1, which can be stated as e ⊤ A ( u ( A ) ◦ v ( A )) ≥ u ( A ) ⊤ A v ( A ) = s ( A ) , with equality only when e ⊤ A = s ( A ) e ⊤ , where u ( A ) ⊤ and v ( A ) are the left and right eigenvectors of A associ-ated with the Perron root s ( A ), and u ◦ v is the compo-nentwise (Schur-Hadamard) product. Without the equal-ity condition, their result is a special case of [50, Theorem3.2.5], but to obtain the equality condition requires anapproach their novel proof provides. Lee Altenberg Remark 3 Schreiber and Lloyd-Smith [51, Appendix B,Lemma 1] followed the reverse path and extended Kirklandet al’s result on s ( M ( m ) D ) to the form s ( α A + D ), where A is essentially nonnegative and D any diagonal matrix.Lemma 1 can also be used as a new proof of an in-equality of Lindqvist, the special case considered in [52,Theorem 2, pp. 260–261]. Theorem 6 (Lindqvist [52, Theorem 2, subcase]) Let A be an irreducible n × n real matrix such that 1) A ij ≥ for i = j , and 2) The left and right eigenvectorsof A , u ( A ) ⊤ and v ( A ) , associated with eigenvalue s ( A ) ,satisfy u ( A ) ⊤ v ( A ) = 1 . Let D be an n × n real diagonalmatrix. Then s ( A + D ) − s ( A ) ≥ u ( A ) ⊤ D v ( A ) . (13) Proof: Since A is an essentially nonnegative matrix, s ( A )is an eigenvalue of multiplicity 1. Consider the representa-tion A = m B − D , where B is essentially nonnegative and m > 0. Write s ≡ s ( A ). As A is irreducible, u ≡ u ( A ), v ≡ v ( A ), with u ⊤ v = 1, e ⊤ v = 1, are unique, and thederivatives exist [26] in the following [53, Sec. 9.1.1]: u ⊤ ∂ ( Av ) ∂m = u ⊤ (cid:18) ∂ A ∂m v + A ∂ v ∂m (cid:19) = u ⊤ Bv + s u ⊤ ∂ v ∂m = u ⊤ ∂∂m ( s v ) = u ⊤ (cid:18) ∂s∂m v + s ∂ v ∂m (cid:19) = ∂s∂m + s u ⊤ ∂ v ∂m . Cancellation of terms s u ⊤ ∂ v /∂m gives ∂s ( A ) ∂m = u ( A ) ⊤ ∂ A ∂m v ( A ) = u ( A ) ⊤ Bv ( A ) ≤ s ( B ) , the inequality coming from Lemma 1. Scaling by m , sub-tracting D , and substituting m B = A + D , we get u ( A ) ⊤ ( m B − D ) v ( A ) = s ( A ) ≤ s ( m B ) − u ( A ) ⊤ Dv ( A ) ⇐⇒ u ( A ) ⊤ Dv ( A ) ≤ s ( A + D ) − s ( A ) . 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