Central diagonal sections of the n -cube
CCENTRAL DIAGONAL SECTIONS OF THE n -CUBE F. A. BARTHA , F. FODOR , AND B. GONZALEZ MERINO Abstract.
We prove that the volume of central hyperplane sections of a unit cube in R n orthog-onal to a diameter of the cube is a strictly monotonically increasing function of the dimension for n ≥
3. Our argument uses an integral formula that goes back to P´olya [P´ol13] (see also [Hen79]and [Bal86]) for the volume of central sections of the cube, and Laplace’s method to estimate theasymptotic behaviour of the integral. First we show that monotonicity holds starting from somespecific n . Then, using interval arithmetic (IA) and automatic differentiation (AD), we computean explicit bound for n , and check the remaining cases between 3 and n by direct computation. Introduction
Let C n = [ − , ] n be the unit cube in R n , and for u ∈ R n let H ( u ) = u ⊥ , the hyperplanethrough o orthogonal to u . We are interested in determining Vol n − ( C n ∩ H ( u )) in the specialcase when u = (1 , . . . , ∈ R n is parallel to a main diagonal of C n .Hensley [Hen79] described a probabilistic argument, whose origin he attributed to Selberg, prov-ing that Vol n − ( C n ∩ H ( u )) → (cid:112) /π as n → ∞ , and he conjectured that max u Vol n − ( C n ∩ H ( u )) ≤ √
2. This conjecture was proved by Ball [Bal86], who proved a integral formula for thevolume of sections that goes back to P´olya [P´ol13], which, when specialized to the case of H ( u ),is the following:(1) I ( n ) := Vol n − ( C n ∩ H ( u )) = 2 √ nπ (cid:90) + ∞ (cid:18) sin tt (cid:19) n dt. It is an interesting fact that the maximum volume hyperplane section of the cube occurs when thehyperplane is orthogonal to u = (1 , , , . . . , (cid:112) /π for the main diagonal is slightly less than √ n − ( C n ∩ H ( u )) = √ n n ( n − n (cid:88) i =0 ( − i (cid:18) ni (cid:19) ( n − i ) n − sign( n − i ) , Date : June 11, 2020.
Key words and phrases.
Cube, sections, volume. Supported by NKFIH KKP 129877 and EFOP-3.6.2-16-2017-00015 grants and by the grant TUDFO/47138-1/2019-ITM of the Ministry for Innovation and Technology, Hungary. Supported by grant TUDFO/47138-1/2019-ITM of the Ministry for Innovation and Technology, Hungary, and byHungarian National Research, Development and Innovation Office NKFIH grant KF129630. This research is a result of the activity developed within the framework of the Programme in Support ofExcellence Groups of the Regi´on de Murcia, Spain, by Fundaci´on S´eneca, Science and Technology Agency of theRegi´on de Murcia. Partially supported by Fundaci´on S´eneca project 19901/GERM/15, Spain, and by MICINNProject PGC2018-094215-B-I00 Spain. a r X i v : . [ m a t h . M G ] J un F. A. BARTHA , F. FODOR , AND B. GONZALEZ MERINO see Goddard [God45], Grimsey [Gri45], Butler [But60], and Frank and Riede [FR12]. Numericalcomputations with (2) suggest that Vol n − ( C n ∩ H ( u )) is a strictly monotonically increasingfunction of n while it tends to the limit (cid:112) /π as n → ∞ . However, (2) does not seem to lend itselfas a tool for proving this monotone property.
20 40 60 80 1001.301.321.341.361.38
Figure 1.
Vol n − ( C n ∩ H ( u )) for 3 ≤ n ≤
110 plotted by
Mathematica .Recently, K¨onig and Koldobsky proved that, in fact, Vol n − ( C n ∩ H ) ≤ (cid:112) /π for all n ≥
2, see[KK19, Prop. 6(a)]. We also point out the recent result of Aliev [Ali20] (see also [Ali08]) abouthyperplane sections of the cube, in which he proves that(3) √ n √ n + 1 ≤ I ( n + 1) I ( n )which is slightly less than the monotonicity of Vol n − ( C n ∩ H ( u )).For a more detailed overview of the currently known information on sections of the cube and forfurther references, see, for example, the books of Berger [Ber10] and Zong [Zon06], and the papersby Ball [Bal86, Bal89], K¨onig, Koldobsky [KK19] and Ivanov, Tsiutsiurupa [IT20].Our main result is the following. Theorem 1.
The volume Vol n − ( C n ∩ H ( u )) is a strictly monotonically increasing function of n for all n ≥ . Theorem 1 directly yields the following corollary (which has already been proved by K¨onig andKoldobsky [KK19]), and slightly improves the estimate (3) of Aliev mentioned above.
Corollary 1.
For any integer n ≥ , it holds thatVol n − ( C n ∩ H ( u )) < (cid:114) π . and this upper bound is best possible. The rest of the paper is organized as follows. In Section 2 we use Laplace’s method to studythe behaviour of the integral (1), and prove the existence of an integer n with the propertythat Vol n − ( C n ∩ H ( u )) is an increasing sequence for all n ≥ n . In the Appendix, using intervalarithmetic, automatic differentiation, and some analytical arguments, we provide rigorous numerical ENTRAL DIAGONAL SECTIONS OF THE n -CUBE 3 estimates, which we use in Section 3 to obtain an explicit upper bound on n . Finally, we checkmonotonicity for 3 ≤ n ≤ n by calculating the value of of Vol n − ( C n ∩ H ( u )) using (2), thusconcluding the proof of Theorem 1.2. Proof of the monotonicity for large n In this section, we prove the following statement which is the most important ingredient of theproof of Theorem 1.
Theorem 2.
There exists an integer n such that Vol n − ( C n ∩ H ( u )) is a strictly monotonicallyincreasing function of n for all n ≥ n .Proof. We are going to examine the behaviour of the integral: I ( n ) = 2 √ nπ (cid:90) + ∞ (cid:18) sin tt (cid:19) n dt, n ≥ . We wish to prove that there exists an n such that I ( n ) is strictly monotonically increasing for all n ≥ n .We start the argument by restricting the domain of integration to a finite interval that containsmost of the integral as n → ∞ . If a fixed, with 1 < a < π/
2, then for n ≥ √ nπ (cid:90) + ∞ a (cid:12)(cid:12)(cid:12)(cid:12) sin tt (cid:12)(cid:12)(cid:12)(cid:12) n dt < √ nπ (cid:90) + ∞ a t − n dt = 2 √ nπ a − n +1 n − < a − n =: e ( n ) . Note that the function e ( n ) tends to 0 exponentially fast as n → + ∞ . Let a be fixed, say, a = 1 .
1, and define(4) I a ( n ) := 2 √ nπ (cid:90) a (cid:18) sin tt (cid:19) n dt, for n ≥ . Then | I ( n ) − I a ( n ) | < e ( n ) for n ≥ . We will use Laplace’s method to study the behaviour of I a ( n ). Let us make the following changeof variables sin tt = e − x / , thus x = (cid:114) − tt , where we define the value of sin t/t to be 1 at t = 0. Therefore, x ( t ) is analytic in the interval [0 , a ].Note that x (0) = 0, and x (cid:48) ( t ) > t ∈ [0 , t ]. Thus, x ( t ) maps [0 , a ] bijectively onto [0 , x ( a )],and so it has an inverse t ( x ) : [0 , x ( a )] → [0 , a ], which is also analytic in [0 , x ( a )] by the LagrangeInversion Theorem because x (cid:48) ( t ) (cid:54) = 0 for t ∈ [0 , a ]. In our case, 1 . < x ( a ) = x (1 .
1) = 1 . < . x ( t ) around t = 0 begins with the terms x = t + t
60 + 139 t t . . . . We can get the first few terms of the the Taylor series expansion of t = t ( x ) around x = 0 byinverting the Taylor series of x ( t ) at t = 0 as follows: t ( x ) = x − x − x x . . . . F. A. BARTHA , F. FODOR , AND B. GONZALEZ MERINO Then t (cid:48) ( x ) = 1 − x − x R ( x )is the order 5 Taylor polynomial of t (cid:48) ( x ) around x = 0 (observe that the degree 5 term is zero), andfor the Lagrange remainder term R ( x ), it holds that R ( x ) = t (7) ( ξ )6! x for some ξ ∈ (0 , x ) (depending on x ). Since t ( x ) is analytic in [0 , x ( a )], in particular the seventhderivative of t ( x ) is analytic too, and thus it is a continuous function. Then the Extreme ValueTheorem yields that t (7) attains its maximum in [0 , x ( a )], and thus | t (7) ( x ) | ≤ R , for some R > x ∈ [0 , x ( a )]. Then we can use the following estimate on x ∈ [0 , x ( a )]:(5) | R ( x ) | ≤ R x , Therefore, after the change of variables, we need to evaluate I a ( n ) = 2 √ nπ (cid:90) x ( a )0 e − nx / t (cid:48) ( x ) dx = 2 √ nπ (cid:90) x ( a )0 e − nx / (cid:18) − x − x R ( x ) (cid:19) dx = 2 √ nπ (cid:90) x ( a )0 e − nx / (cid:18) − x − x (cid:19) dx + 2 √ nπ (cid:90) x ( a )0 e − nx / R ( x ) dx. In order to calculate the above integrals we will use the central moments of the normal distribu-tion: If y = √ πσ e − ( x − µ )22 σ , then for an integer p ≥ E [ y p ] = (cid:40) , if p is odd ,σ p ( p − , if p is even . In our case σ = 3 /n . Thus, using (6) and (5), we get that2 √ nπ (cid:90) x ( a )0 e − nx / | R ( x ) | dx ≤ R √ nπ (cid:90) x ( a )0 e − nx / x dx< R √ nπ (cid:90) + ∞ e − nx / x dx = 2 R √ nπ
6! 3 n R π n / < R n / =: e ( n ) . Notice also that 2 √ nπ (cid:90) + ∞ e − nx / (cid:18) − x − x (cid:19) dx ENTRAL DIAGONAL SECTIONS OF THE n -CUBE 5 = (cid:114) π √ nπ (cid:18) n / − n / − n / (cid:19) = (cid:114) π (cid:18) − n − n (cid:19) . The complementary error function is defined aserfc( x ) := 2 1 √ π (cid:90) + ∞ x e − τ dτ. It is known that erfc( x ) ≤ e − x for x ≥
0. Then, taking into accout that x ( a ) > .
07, we obtain2 √ nπ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) + ∞ x ( a ) e − nx / (cid:18) − x − x (cid:19) dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ √ nπ (cid:90) + ∞ x ( a ) e − nx / (cid:12)(cid:12)(cid:12)(cid:12) − x − x (cid:12)(cid:12)(cid:12)(cid:12) dx ≤ √ nπ (cid:90) + ∞ x ( a ) e − nx / (cid:18) x
20 + 13 x (cid:19) dx< √ nπ (cid:90) + ∞ e − nx / (cid:18) x
20 + 13 x (cid:19) dx = (cid:114) π erfc( (cid:112) n/ (cid:18)
13 + 168 n + 1120 n n (cid:19) + 2 e − n/ √ n
117 + 1525 n πn < e − n/ =: e ( n ) . Now, using the monotonicity of e ( n ), we obtain that I ( n + 1) − I ( n ) ≥ ( I a ( n + 1) − e ( n + 1)) − ( I a ( n ) + e ( n )) ≥ I a ( n + 1) − I a ( n ) − e ( n ) . Furthermore, I a ( n + 1) ≥ (cid:114) π (cid:18) − n + 1) − n + 1) (cid:19) − e ( n + 1) − e ( n + 1) , and I a ( n ) ≤ (cid:114) π (cid:18) − n − n (cid:19) + e ( n ) + e ( n ) . Therefore I ( n + 1) − I ( n ) ≥ (cid:114) π (cid:18) n − n + 1) + 131120 n − n + 1) (cid:19) − e ( n ) − e ( n ) − e ( n + 1) − e ( n ) − e ( n + 1)= (cid:114) π (cid:18) n ( n + 1) + 13(2 n + 1)1120 n ( n + 1) (cid:19) − a − n − ( e ( n ) + e ( n + 1) + e ( n ) + e ( n + 1)) > (cid:114) π (cid:18) n ( n + 1) (cid:19) − a − n − e ( n ) − e ( n ) F. A. BARTHA , F. FODOR , AND B. GONZALEZ MERINO > (cid:114) π (cid:18) n ( n + 1) (cid:19) − a − n − Rn / − e − n/ ≥ (cid:114) π (cid:18) n ( n + 1) (cid:19) − · . − n − Rn / − e − n/ . (7)Clearly, there exists an n , such that for all n ≥ n the expression (7) is strictly positive. Thus,Vol n − ( C n ∩ H ( u )) is strictly monotonically increasing for n ≥ n .Thus, we have finished the proof of Theorem 2. (cid:3) Remark.
Figure 1 suggests that Vol n − ( C n ∩ H ( u )) is not only a monotonically increasing sequencebut also concave, i.e., 2 I ( n + 1) ≥ I ( n ) + I ( n + 2) for n ≥
3. We note, without giving the details,that with a similar argument as in the proof of Theorem 2, but using more terms of the Taylorexpansion of t ( x ), one can also show that2 I ( n + 1) − I ( n ) − I ( n + 2) ≥ I a ( n + 1) − I a ( n ) − I a ( n + 2) − ξ e ( n ) − ξ e ( n ) − ξ e ( n ) ≥ √ √ π n ( n + 1)( n + 2) + O ( n − ) − ξ e ( n ) − ξ e ( n ) − ξ e ( n ) , for some ξ i > i = 1 , ,
3. If we take into account sufficiently many terms of the Taylor series of t ( x ), then we can guarantee that each error term is of smaller order than n − , and thus there existsa number n such that the sequence I ( n ) is concave for all n ≥ n .3. Proof of Theorem 1
In order to prove Theorem 1, we need an explicit upper bound on the critical number n . Usinga combination of interval arithmetic, automatic differentiation, and some analytic methods, we canobtain a rigorous upper estimate for the seventh derivative | t (7) ( x ) | in x ∈ [0 , x ( a )]. We providethe details of this argument in the Appendix. Here, we only quote the following upper bound (seeTheorem 3 part (3)):(8) R ≤ . . Now, substituting the estimate (8) in inequality (7), we get that n < I ( n + 1) − I ( n ) using (2) to the required accuracy, and verify monotonicity for all3 ≤ n ≤ n n Figure 2. I ( n + 1) − I ( n ) for 3 ≤ n ≤
145 plotted by
MathematicaRemark.
Using the same ideas as above, one could show the concavity of I ( n ) for n ≥ ENTRAL DIAGONAL SECTIONS OF THE n -CUBE 7 Acknowledgements
We would like to thank Juan Arias de Reyna for helpful discussions and suggestions.
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On the volume of hyperplane sectionf of a d -cube , arXiv:2004.00873v1 (2020).[Ali08] I. Aliev, Siegels Lemma and Sum-Distinct Sets , Discrete Comput. Geom. (2008), no. 3, 59–66.[Bal86] K. Ball, Cube slicing in R n , Proc. Amer. Math. Soc. (1986), no. 3, 465–473.[Bal89] K. Ball, Volumes of sections of cubes and related problems , Geometric aspects of functional analysis (1987–88), Lecture Notes in Math., vol. 1376, Springer, Berlin, 1989, pp. 251–260.[Ber10] M. Berger,
Geometry revealed , Springer, Heidelberg, 2010. A Jacob’s ladder to modern higher geometry;Translated from the French by Lester Senechal.[But60] R. Butler,
On the evaluation of (cid:82) ∞ (sin m t ) /t n dt by the trapezoidal rule , Amer. Math. Monthly (1960),566–569.[FR12] R. Frank and H. Riede, Hyperplane sections of the n -dimensional cube , Amer. Math. Monthly (2012),no. 10, 868–872.[Hen79] D. Hensley, Slicing the cube in R n and probability (bounds for the measure of a central cube slice in R n by probability methods) , Proc. Amer. Math. Soc. (1979), no. 1, 95–100.[God45] L. S. Goddard, LII. The accumulation of chance effects and the Gaussian frequency distribution , TheLondon, Edinburgh, and Dublin Philosophical Magazine and Journal of Science (1945), no. 257, 428–433.[Gri45] A. H. R. Grimsey, XL. On the accumulation of chance effects and the Gaussian Frequency Distribution:To the editors of the Philosophical Magazine , The London, Edinburgh, and Dublin Philosophical Magazineand Journal of Science (1945), no. 255, 294–295.[IT20] G. Ivanov and I. Tsiutsiurupa, On the volume of sectons of the cube , arXiv:2004.02674 (2020).[KK19] H. K¨onig and A. Koldobsky,
On the maximal perimeter of sections of the cube , Adv. Math. (2019),773–804.[P´ol13] G. P´olya,
Berechnung eines bestimmten Integrals. , Math. Ann. (1913), 204–212.[Zon06] C. Zong, The cube: a window to convex and discrete geometry , Cambridge Tracts in Mathematics, vol. 168,Cambridge University Press, Cambridge, 2006.
Appendix
Consider the function(9) x ( t ) = (cid:115) − (cid:18) sin tt (cid:19) , where t ∈ [0 , . sin tt is understood to be augmented with its limit at t = 0 that is sin 00 = 1. Then, the function x ( t ) is analytic. Theorem 3.
The following holds true. (1) x ( t ) is strictly increasing on [0 , . and x ( t ) ≤ . for t ∈ [0 , . . (2) x ( t ) is invertible with inverse t ( x ) , where x ∈ [0 , x (1 . . (3) The 7th derivative of t ( x ) attains the upper bound (cid:12)(cid:12)(cid:12) t (7) ( x ) (cid:12)(cid:12)(cid:12) ≤ . for x ∈ [0 , x (1 . . F. A. BARTHA , F. FODOR , AND B. GONZALEZ MERINO The monotonicity stated in (1) is trivial, hence, one just needs to establish the containment x (1 . ∈ [0 , x (1 . x ( t ) and proving (3).There are numerous computational steps involved. In order to obtain rigorous results, we havebased our computations on two techniques, namely, interval arithmetic (IA) and automatic differ-entiation (AD) that are capable of providing mathematically sound bounds for functions and theirderivatives alike. Besides the technical hurdle, severe difficulties arise at the left endpoint t = 0 as,when computing the derivatives of x ( t ), we need to differentiate both √· and sin tt at zero. It wastempting to use Taylor models, an advanced combination of these two, however that could still nothandle the aforementioned left endpoint directly, hence, we chose to stick with the straightforwardapplication of the two techniques and used the CAPD package [1]. For a comprehensive overviewof these topics we refer to [2, 3, 4].We emphasize that the major goal of Theorem 3 is providing the given bounds, hence, we madelittle effort to obtain tighter results and were performing sub–optimal computations knowingly, inorder to decrease the implementation burden.The key step to overcome the difficulties at t = 0 is to rephrase (9) as x ( t ) = t (cid:112) h ( t ) ,h ( t ) = ( g ◦ F )( t ) · ( − F ( t )) ,g ( t ) = log(1 + t ) t ,F ( t ) = t F ( t ) , and F ( t ) = sin( t ) t − t . (10)Section 5 details the considerations used for dealing with the functions appearing in (10). Inparticular, Sections 5.1 and 5.2 handle the functions sin tt and F ( t ); a computational scheme fortheir derivatives is provided. Then, we turn our attention to log(1+ t ) t and derive analogous resultsin Section 5.3. The square root is discussed in Section 5.4. Then, in Section 5.5, we present apure formula for the higher order chain–rule used to compose g ( t ) and F ( t ). Section 6 contains theresults for x ( t ) and its derivatives, in particular, the proof of the remaining part of (1) in Theorem 3.Section 7 deals with t ( x ) by giving a general inversion procedure in Section 7.1 and the final proofin Section 7.2.The codes performing the rigorous computational procedure described in this manuscript, to-gether with the produced outputs, are publicly available at [6].5. Bounding functions and their derivatives
First, we will closely analyze some Taylor expansions centered at t = 0 and derive bounds forTaylor coefficients of the very same functions expanded around another center point ˆ t . Then, weinclude the higher order chain–rule for completeness.5.1. The function sin tt . The Taylor series of sin t centered at t = 0 is given assin t = ∞ (cid:88) k =0 ( − k t k +1 (2 k + 1)! ENTRAL DIAGONAL SECTIONS OF THE n -CUBE 9 and is convergent for all t ∈ R . Consequently, we obtain the Taylor series of f ( t ) := (cid:40) sin tt , if t > , , if t = 0as(11) f ( t ) = ∞ (cid:88) k =0 ( − k t k (2 k + 1)! , again, centered at t = 0 with the same convergence radius. Therefore,(12) 1 m ! f ( m ) ( t ) = 1 m ! ∞ (cid:88) k ≥ m/ ( − k t k − m (2 k − m )! · (2 k + 1) for m = 0 , , . . . We shall bound these infinite series as follows. Let N ≥ m/
2, then, define the finite part S f ( t ; N, m )and the remainder part E f ( t ; N, m ) as S f ( t ; N, m ) = 1 m ! N (cid:88) k ≥ m/ ( − k t k − m (2 k − m )! · (2 k + 1) and E f ( t ; N, m ) = 1 m ! ∞ (cid:88) k = N +1 ( − k t k − m (2 k − m )! · (2 k + 1) . (13)The following lemma establishes bounds for the remainder. Lemma 1.
Let m, N ∈ Z with m ≥ and N ≥ m/ . Then, E f ( t ; N, m ) ∈ m ! e t (2 N + 2 − m )! t N +2 − m · [ − , for all t ≥ .Proof. Let t ≥ | E f ( t ; N, m ) | ≤ m ! ∞ (cid:88) k = N +1 t k − m (2 k − m )! · (2 k + 1) ≤ m ! ∞ (cid:88) k = N +1 t k − m (2 k − m )! ≤ m ! ∞ (cid:88) k =2 N +2 − m t k k ! . Note that we have obtained the tail of the Taylor series of the exponential function centered at t = 0. The corresponding Lagrange remainder gives us that for all integers K ≥ ∞ (cid:88) k = K t k k ! = 1 K ! e t ( K ) ( ξ ) · t K holds with some ξ ≡ ξ ( K ) ∈ [0 , t ]. Observe that e t ( K ) ( ξ ) = e ξ and that attains its maximum at ξ = t over ξ ∈ [0 , t ]. Hence, we obtain ∞ (cid:88) k = K t k k ! ≤ K ! e t · t K for t ≥ . Finally, setting K = 2 N + 2 − m and deriving a bound on E f ( t ; N, m ) from the estimate for | E f ( t ; N, m ) | concludes the proof. (cid:3) , F. FODOR , AND B. GONZALEZ MERINO Defining(14) E f ( t ; N, m ) = 1 m ! e t (2 N + 2 − m )! t N +2 − m · [ − , f ( t ) and itsderivatives, namely, 1 m ! f ( m ) ( t ) ∈ S f ( t ; N, m ) + E f ( t ; N, m ) . We remark that lim N →∞ E f ( t ; N, m ) → { } for all t ∈ R and m ≥
0. Figure 3 gives an insight onhow the obtained bound for the remainder behaves. - - -
10 1 1.11e - - - E f ( t, 5, 0 ) E f ( t, 10, 0 ) E f ( t, 15, 0 ) E f ( t, 20, 0 ) - - -
10 1 1.11e - - - E f ( t, 5, 1 ) E f ( t, 10, 1 ) E f ( t, 15, 1 ) E f ( t, 20, 1 ) - - -
10 1 1.11e - - - E f ( t, 5, 4 ) E f ( t, 10, 4 ) E f ( t, 15, 4 ) E f ( t, 20, 4 ) - - -
10 1 1.11e - - - E f ( t, 5, 7 ) E f ( t, 10, 7 ) E f ( t, 15, 7 ) E f ( t, 20, 7 ) Figure 3.
The upper bound of E f ( t ; N, m ) for various (
N, m ) over t ∈ [0 , . The function t F ( t ) = sin tt . Even though there are no issues with directly computinglog ( f ( t )) using the results above, as shown in (10), we will need a more sophisticated approach inorder to be able to tackle the final square root operation in the neighbourhood of zero. To thatend, we rewrite expansion (11) as f ( t ) = 1 + t F ( t ) = 1 + t ∞ (cid:88) k =0 ( − k +1 t k (2 k + 3)! , ENTRAL DIAGONAL SECTIONS OF THE n -CUBE 11 a 2nd–order Taylor model. Analogous arguments, as in Section 5.1, lead to the following. Lemma 2. m ! F ( m ) ( t ) ∈ S F ( t ; N, m ) + E F ( t ; N, m ) , where S F ( t ; N, m ) = 1 m ! N (cid:88) k ≥ m/ ( − k +1 t k − m (2 k − m )! · (2 k + 1)(2 k + 2)(2 k + 3) and E F ( t ; N, m ) = E f ( t ; N, m ) . Note that the remainder bound is identical to the one in (14) as the factor in the denominator(2 k + 1)(2 k + 2)(2 k + 3) may be eliminated the same way as (2 k + 1) in the proof of Lemma 1.5.3. The function log(1+ t ) t . Following (10), we will compute x ( t ) using the form x ( t ) = t (cid:115) − F ( t ) log (1 + t F ( t )) t F ( t ) . Thus, the next step is to analyze g ( t ) = log(1+ t ) t , where t ∈ ( − , t ). At t = 0, we augment with the limit g (0) := 1. Note that theargument of g ( · ) will be t F ( t ) = sin tt − − . , t ) centered at t = 0, namely,log(1 + t ) = ∞ (cid:88) k =1 ( − k +1 t k k that is convergent for | t | <
1. Then, formally, g ( t ) = ∞ (cid:88) k =0 ( − k t k k + 1and 1 m ! g ( m ) ( t ) = 1 m ! ( − m ∞ (cid:88) k =0 ( − k t k k + m + 1 ( k + m )! k ! for m = 0 , , . . . follow that may be simplified as1 m ! g ( m ) ( t ) = ( − m ∞ (cid:88) k =0 ( − k (cid:18) k + mm (cid:19) t k k + m + 1 . We define S g ( t ; N, m ) = ( − m N (cid:88) k =0 ( − k (cid:18) k + mm (cid:19) t k k + m + 1 and E g ( t ; N, m ) = ( − m ∞ (cid:88) k = N +1 ( − k (cid:18) k + mm (cid:19) t k k + m + 1(15)for N ≥ N ≥ m so that k + m > m in the binomial coefficients in E g ). We maybound the remainder part as detailed below. , F. FODOR , AND B. GONZALEZ MERINO Lemma 3.
Let N ≥ m ≥ and t ∈ ( − , . Then, | E g ( t ; N, m ) | ≤ (cid:40) | t | N +1 (1 −| t | ) N +2 , if m = 0 , (cid:0) em (cid:1) m m ! (cid:0) m + N +1 m (cid:1) | t | N +1 (1 −| t | ) m + N +2 , else . Proof.
When m = 0, the binomial coefficient (cid:0) k + mm (cid:1) = 1, thus, | E g ( t ; N, | ≤ ∞ (cid:88) k = N +1 | t | k k + 1 ≤ ∞ (cid:88) k = N +1 | t | k . On the other hand, for m >
0, it is known that (cid:18) k + mm (cid:19) ≤ (cid:18) e ( k + m ) m (cid:19) m . Hence, | E g ( t ; N, m ) | ≤ ∞ (cid:88) k = N +1 (cid:18) e ( k + m ) m (cid:19) m | t | k k + m + 1 ≤ ∞ (cid:88) k = N +1 (cid:18) e (1 + mk ) m (cid:19) m k m | t | k k + m + 1 ≤ (cid:18) em (cid:19) m ∞ (cid:88) k = N +1 k m | t | k . Thus, for both cases, it is sufficient to bound the series ∞ (cid:88) k = N +1 k m | t | k for all m = 0 , , . . . , N ≥ m and | t | < T = | t | ∈ [0 ,
1) and consider the m -th derivative of theconvergent geometric series 11 − T = ∞ (cid:88) k =0 T k that is (cid:18) − T (cid:19) ( m ) = ∞ (cid:88) k =0 ( k + m )! k ! T k . We may easily bound the remainder of this series starting from k = N +1 using, again, the Lagrangeformula as ∞ (cid:88) k = N +1 ( k + m )! k ! T k = ( m + N +1) − T (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) T = ξ · N + 1)! · T N +1 with some ξ ∈ [0 , T ]. The K -th derivative of − T = (1 − T ) − is given by K ! (1 − T ) − ( K +1) thatis clearly maximal for ξ = T . Hence, ∞ (cid:88) k = N +1 ( k + m )! k ! T k ≤ ( m + N + 1)! (1 − T ) − ( m + N +2) N + 1)! T N +1 that concludes the proof by noting that ∞ (cid:88) k = N +1 k m T k ≤ ∞ (cid:88) k = N +1 ( k + m )! k ! T k ENTRAL DIAGONAL SECTIONS OF THE n -CUBE 13 holds for all N ≥ m ≥ T ∈ [0 , (cid:3) In summary, letting E g ( t ; N, m ) := [ − , · (cid:40) | t | N +1 (1 −| t | ) N +2 , if m = 0 , (cid:0) em (cid:1) m m ! (cid:0) m + N +1 m (cid:1) | t | N +1 (1 −| t | ) m + N +2 , else , provides the computational method(16) 1 m ! g ( m ) ( t ) ∈ S g ( t ; N, m ) + E g ( t ; N, m ) . To analyze the dynamics of (16), observe that the behaviour of the remainder is governed by (cid:18) m + N + 1 m (cid:19) (cid:18) | t | − | t | (cid:19) N for fixed t ∈ ( − ,
1) and m ≥
0. Using the same bound as above for the binomial, it is easy to seethat, eventually, N m (cid:18) | t | − | t | (cid:19) N determines the limit, when N → ∞ . Therefore,lim N → ∞ E g ( t ; N, m ) = { } , when | t | −| t | < | t | < . Recall that for our case this will be satisfied as sin 1 . . − ≈ − . E g ( t ; N, m ) is demonstrated on Figure 4.5.4.
The function (cid:112) t h ( t ) . Assume t ∈ [0 , T ] with some T ≥
0. By itself, the function √ t is notdifferentiable at t = 0. However, if the argument is of the special form t h ( t ) with h ( t ) (cid:54) = 0, thenthe situation changes as (cid:112) t h ( t ) = t (cid:112) h ( t ) , hence, () (cid:112) t h ( t ) = (cid:112) h ( t ) + t h (cid:48) ( t )2 (cid:112) h ( t )implying no difficulties for all t ∈ [0 , T ].5.5. The chain–rule.
There are numerous known formulae for the higher order chain rule [5]. Weshall use the classical one named after Fa di Bruno that is written as follows.
Lemma 4 (Fa di Bruno) . Let f : I → U and g : U → V be analytic functions, where I, U, V ⊆ R are connected subsets. Consider the Taylor expansions f ( t ) = (cid:80) ∞ k =0 ( f ) k ( t − t ) k centered at t ∈ I with t ∈ I and g ( x ) = (cid:80) ∞ k =0 ( g ) k ( x − x ) k centered at x = f ( t ) for x ∈ U . Then, the compositefunction ( g ◦ f ) attains the Taylor expansion ( g ◦ f )( t ) = (cid:80) ∞ k =0 ( g ◦ f ) k ( t − t ) k centered at t withthe coefficients ( g ◦ f ) = ( g ) and ( g ◦ f ) k = (cid:88) b +2 b + ... + kb k = km := b + b + ... + b k m ! b ! b ! . . . b k ! ( g ) m k (cid:89) i =1 (cid:16) ( f ) i (cid:17) b i , (17) where k ≥ and b , . . . , b k are nonnegative integers. , F. FODOR , AND B. GONZALEZ MERINO - - -
10 0.2 0.3 0.41e - - - E g ( t, 5, 0 ) E g ( t, 10, 0 ) E g ( t, 15, 0 ) E g ( t, 20, 0 ) - - -
10 0.2 0.3 0.41e - - - E g ( t, 5, 1 ) E g ( t, 10, 1 ) E g ( t, 15, 1 ) E g ( t, 20, 1 ) - - -
10 0.2 0.3 0.41e - - - E g ( t, 5, 4 ) E g ( t, 10, 4 ) E g ( t, 15, 4 ) E g ( t, 20, 4 ) - - -
101 0.2 0.3 0.41e - - - E g ( t, 5, 7 ) E g ( t, 10, 7 ) E g ( t, 15, 7 ) E g ( t, 20, 7 ) Figure 4.
The upper bound of E g ( t ; N, m ) for various values.Note that we altered the notation somewhat compared to [5] and use Taylor coefficients insteadof derivatives, this should not cause confusion.6.
Derivatives of x ( t )Using the combination of results of Section 5, we may attempt to evaluate x ( t ) and its derivativesbased on the steps detailed in (10). The expansions of − t , and t are trivial, so is the applicationof the product rule; for the square root, the computation of Taylor coefficients is straightforward[2, 3].We used a uniform N = 20 when executing our program and imposed 0 (cid:54)∈ x (1) ([0 , . , .
1] into smaller intervals so that each was no longer than ≈ . x ( t ) directly from (9) as well.This clearly failed for those close to t = 0, however, whenever it succeeded, we compared it withthe results from scheme (10) and used the intersection of the two, somewhat independent, results.The obtained enclosures are given in Table 1. Each row presents the interval hull of the rigorousbounds obtained over all small subintervals. In particular, the first one establishes the remainingpart of (1) in Theorem 3. ENTRAL DIAGONAL SECTIONS OF THE n -CUBE 15 Taylor coefficient for any t ∈ [0 , . is contained in ( x ) [0 , . x ) [0 . , . x ) [ − . − , . x ) [0 . , . x ) [ − . − , . x ) [0 . , . x ) [ − . − , . x ) [6 . − , . Table 1.
Bounds on Taylor coefficients of x ( t ) centered at t ∈ [0 , . . Derivatives of t ( x )Now, that we have computed rigorous bounds for the Taylor coefficients of x ( t ) up to the desiredorder for any center t ∈ [0 , . t ( x ). First, in Section 7.1, wepresent the general formula for computing the inverse expansion, then, we include the results of ourcomputation for t ( x ) in Section 7.2, thereby concluding the proof of (3) in Theorem 3.7.1. Derivatives of the inverse function.
Practical formulae for Taylor expansion of the inversefunction based on the coefficients of the original one are rather scarce. For our purposes it isreasonable to utilize the result of Fa di Bruno, seen in Section 5.5, directly.Assume that x ( t ) has the expansion x ( t ) = (cid:80) ∞ k =0 ( x ) k ( t − t ) k centered at t and ( x ) (cid:54) = 0. Then,for the inverse we may construct the expansion t ( x ) = (cid:80) ∞ k =0 ( t ) k ( x − x ) k centered at x = x ( t )as ( t ) = t , ( t ) = 1( x ) , and( t ) k = − (cid:88) b +2 b + ... + kb k = km := b + b + ... + b k m (cid:54) = k m ! b ! b ! . . . b k ! ( t ) m (cid:16) ( x ) (cid:17) b − k k (cid:89) i =2 (cid:16) ( x ) i (cid:17) b i , (18)for k ≥
2. The first two coefficients are trivial. The general part is a consequence of Lemma 4applied to ( t ◦ x )( t ) by observing that ( t ◦ x ) k = 0 for k ≥ t ) k (that is ( g ) k in the original Lemma) is given by b = k and b i = 0 for all other i -sas k ! k ! 1! . . .
1! ( t ) k (cid:16) ( x ) (cid:17) k = ( t ) k (cid:16) ( x ) (cid:17) k . Proof of (3) in Theorem 3.
We have applied (18) on each of the subintervals and thecorresponding expansion of x ( t ), see Section 6. The interval hull of the results is presented inTable 2. Using that ( t ) = t (7) ( x ), we directly obtain the claim of (3) in Theorem 3. , F. FODOR , AND B. GONZALEZ MERINO Taylor coefficient for any x ∈ [0 , x (1 . is contained in ( t ) [0 , . t ) [0 . , . t ) [ − . , . − t ) [ − . , − . t ) [ − . , . − t ) [ − . − , . t ) [ − . − , . t ) [ − . − , . − Table 2.
Bounds on Taylor coefficients of t ( x ) centered at x ∈ [0 , x (1 . . References [1] CAPD Group. CAPD Library: Computer Assisted Proofs in Dynamics.
Jagiellonian University . http://capd.ii.uj.edu.pl/index.php .[2] Tucker, W. Validated Numerics: A Short Introduction to Rigorous Computations ; Princeton UniversityPress: Princeton, NJ, USA, 2011. https://doi.org/10.2307/j.ctvcm4g18 .[3] Griewank, A.; Walther, A.
Evaluating Derivatives: Principles and Techniques of Algorithmic Differen-tiation (Second Edition); Society for Industrial and Applied Mathematics (SIAM): Philadelphia, PA,USA, 2008. https://doi.org/10.1137/1.9780898717761 .[4] Makino, K.; Berz, M. Taylor Models and Other Validated Functional Inclusion Methods.
Int.J. Pure Appl. Math. , , 379–456. https://bt.pa.msu.edu//pub/papers/TMIJPAM03/TMIJPAM03.pdf .[5] Johnson, W.P. The Curious History of Fa di Bruno’s Formula. The American Mathematical Monthly , , 217–234. https://doi.org/10.1080/00029890.2002.11919857 .[6] Ferenc A. Bartha. Code: Rigorous Computations. . http://ferenc.barthabrothers.com/math/n-cube.tar.gz . Department of Applied and Numerical Mathematics, University of Szeged, Aradi v´ertan´uk tere 1,6720 Szeged, Hungary
E-mail address : [email protected] Department of Geometry, University of Szeged, Aradi v´ertan´uk tere 1, 6720 Szeged, Hungary
E-mail address : [email protected] Departamento de Did´actica de la Matem´atica, Facultad de Educaci´on, Universidad de Murcia, 30100-Murcia, Spain
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