
328 B.R. 137 (2005)
In re VANTAGE INVESTMENTS, INC., Debtor.
No. 04-046536-11-DRD.
United States Bankruptcy Court, W.D. Missouri.
August 5, 2005.
*138 *139 Erlene W. Krigel, Krigel and Krigel, P.C., Kansas City, MO, for Debtor.

MEMORANDUM OPINION
DENNIS R. DOW, Bankruptcy Judge.
The matter before the Court in this Chapter 11 proceeding is the Request for Allowance and Payment of Administrative Expense ("Request") filed by Best Western International, Inc. ("Best Western"), based on a Membership Agreement ("Agreement") between Best Western and Debtor Vantage Investments, Inc. ("Debtor"). Best Western seeks this Court's order declaring that certain fees, dues and charges owed under the Agreement, liquidated damages based upon the Debtor's *140 continued use of trademarks and logos after the termination of the Agreement and attorney's fees incurred by its counsel in this proceeding should be allowed and paid as expenses of administration. Both the Debtor and its principal secured creditor, University National Bank, N.A. ("Bank") opposed the Request. Debtor contends that it is not obligated to pay those charges accruing subsequent to the date of termination or, in the alternative, that those amounts provide no benefit to the estate and therefore are not allowable as administrative expenses. Debtor also contends that the provision in the Agreement for liquidating damages is unenforceable as a penalty and, in the alternative, that Best Western has incorrectly calculated the amount that might be due pursuant to that provision. This Court has jurisdiction over this matter pursuant to 28 U.S.C. §§ 1334(b) and 157(a) and (b). This is a core proceeding which this Court may hear and determine pursuant to 28 U.S.C. § 157(b)(2)(A). This Memorandum Opinion contains my Findings of Fact and Conclusions of Law pursuant to Rule 52 of the Federal Rules of Civil Procedure as made applicable to this matter by Rules 7052 and 9014(c) of the Federal Rules of Bankruptcy Procedure. For all the reasons set forth below, the Court grants the Request in part and overrules it in part.

I. FACTUAL AND PROCEDURAL BACKGROUND
Debtor owns and operates a hotel known at the time of filing as Best Western KC South. Pursuant to the Agreement with Best Western dated October 7, 2003, Debtor became affiliated with Best Western and was entitled to certain benefits, including usage of certain trademarks and logos of Best Western, participation in the nationwide reservation system and utilization of cooperative purchasing programs. In turn, Debtor was required to pay annual dues and certain monthly fees and charges and also to conform with certain guidelines for its operation and maintenance of the property, including submitting to inspections by Best Western personnel to insure compliance with those guidelines. Debtor's membership was conditionally approved and subject to the requirement that Debtor complete certain items identified in a property improvement plan. The agreement was for an initial term to expire on November 30, 2003, but was renewable thereafter for additional terms ending on November 30 of subsequent years. Debtor had the right to terminate the Agreement at any time. Best Western had the right to terminate the agreement on default by Debtor in performance of any obligation owed to Best Western under the Agreement.
At various times prior to the filing of the petition, Debtor apparently failed to timely pay certain fees and charges and to comply with deadlines established by Best Western for completion of items on the property improvement plan. An agreement was reached by the parties on these issues and embodied in a letter from Best Western to Debtor dated August 30, 2004 in which Best Western agreed to defer cancellation of Debtor's membership subject to certain conditions. One of the conditions was that in the event Debtor became 60 days delinquent in its account to Best Western, its participation in the reservation system would be restricted and its membership automatically cancelled within ten days if the account was not brought current. Another of the requirements was that on a Quality Assurance Assessment, an inspection conducted by Best Western pursuant to the standards agreed upon by the members, Debtor achieve a minimum score of 875 with regard to guest rooms and public areas. On October 25, 2004, Best Western sent Debtor a letter advising *141 that as a result of continued payment default, it had suspended Debtor from participation in the reservation system effective October 18, 2004. The letter also advised that unless a payment (in the amount of $12,179.41, the balance due as of that date) was made by November 4, 2004, the Agreement would be automatically cancelled.
On October 20, 2004, Debtor filed a petition for relief in this Court under Chapter 11 of the Bankruptcy Code. After unsuccessfully attempting to resolve the monetary defaults with Best Western, on November 15, 2004, Debtor filed an Emergency Motion to Assume Executory Contract and to Compel Best Western International, Inc. to Comply With Terms of Contract. In that motion, Debtor sought an order authorizing the assumption of the Agreement with a cure of monetary and other defaults pursuant to a schedule suggested by Debtor and an order requiring Best Western to restore Debtor's ability to participate in the reservation system. Debtor proposed (in an amended motion) to cure the existing monetary default of $18,483.51 by making a payment of $12,000.00 by December 3, 2004, a $5,000 payment by December 31, 2004, a $5,000.00 payment by January 31, 2005 and the balance by February 28, 2005. Debtor also proposed to cure any existing non-monetary defaults within 45 days of the entry of an order authorizing its assumption of the Agreement. After a hearing, this Court entered an Order dated December 2, 2004, later amended on February 7, 2005, authorizing the Debtor to assume the Agreement upon the condition that the monetary and non-monetary defaults be cured within the specified time periods. That Order also provided that in the event the Debtor failed to cure the defaults by the specified dates, Best Western could issue a notice to the Debtor after which the Debtor would have ten days to either cure the defaults or to object by filing a pleading with this Court and seeking a hearing.
In order to determine whether Debtor had cured the non-monetary defaults, Best Western personnel conducted an inspection of Debtor's facility on January 18, 2005. Debtor received a score of 746. Based on this result, Best Western sent a notice of default on February 17, 2005. Best Western conducted a follow-up inspection on March 7, 2005, at which time Debtor's score was 816, still shy of the 875 minimum score required. On March 28, 2005, Best Western sent Debtor a letter advising that as a result of Debtor's failure to achieve the required minimum score, the Agreement would be canceled effective April 7, 2005. On April 6, 2005, Debtor filed an objection to Best Western's notice of default and requested a hearing with this Court to determine whether it had, as it contended, cured the default. After a hearing conducted on April 28, 2005 this Court determined that the Debtor had not cured its non-monetary defaults within the time specified by the Order or within the additional time provided in Best Western's notice of default, that the stay had therefore expired, pursuant to this Court's Order of February 7, 2005, and that Best Western had the right to take action under the Agreement and applicable law to terminate it.
Best Western filed the Request seeking the allowance and payment of administrative expenses in the total amount of $132,140.50. That amount consists of the following: (1) dues, fees and other charges owed under the Agreement, including the balance of the pre-petition defaults and amounts due under the Agreement through the end of November 2005, in the amount of $51,094.16; (2) liquidated damages, pursuant to paragraph 24 of the Agreement, for the Debtor's continued use *142 of Best Western's trademarks and logos subsequent to the date of termination, in the amount of $48,825.00; and (3) attorney's fees incurred by Best Western's counsel relating to Debtor's defaults under the Agreement, pursuant to paragraph 36 of the Agreement, in the amount of $32,221.35. As noted above, Debtor and the Bank have both objected to the request.

II. DISCUSSION AND ANALYSIS

A. Contractual Damages
One component of the Request is a claim for the balance of various dues, fees and charges due under the Agreement. Debtor concedes that the amount of $866.41 is still due and unpaid from monetary defaults existing prior to the date of the filing of the petition and does not object to the allowance and payment of this sum as an administrative expense. As of April 1, 2005, Best Western claims that the sum of $22,951.60 is due for various fees, charges and costs for post-petition services provided to the Debtor pursuant to the Agreement. Likewise, Debtor does not object to these sums.
The balance of Best Western's claim for contractual damages is for monthly fees and charges due pursuant to the Agreement from April 1 to November 30, 2005, the date of the expiration of the term of the Agreement. Debtor and the Bank object to that portion of the claim suggesting that Best Western has no right to amounts due for periods after termination and that if it has such a right, those amounts would not properly be administrative expense claims because no benefit would be conferred on the estate in exchange for those sums.
As to the contractual right, paragraph 11 of the Agreement states that a member must abide by all terms and provisions of the Bylaws. Article II, Section 1(F) of the Bylaws, states that if a member resigns or is terminated, it must still pay dues and fees due for the remainder of the fiscal year; the fiscal year ends November 30[1]. As noted above, November 30 is the anniversary of expiration of the Agreement, which although initially for a term of one year, was subject to renewal. The Court concludes, therefore, that Best Western has a contractual right to the dues and fees claimed for the balance of the term of the Agreement.
Debtor next argues that because the fees and dues are for periods during which Best Western would have rendered no services, there is no benefit to the estate and the claim does not qualify for administrative expense treatment. In general, it is true that in order for a claim to be treated as an expense of administration, it must arise from a transaction with a debtor-in-possession and confer a benefit on the estate. See State of Texas v. Lowe (In re H.L.S. Energy Co., Inc.), 151 F.3d 434, 437 (5th Cir.1998); Ford Motor Credit Co. v. Dobbins, 35 F.3d 860, 866-867 (4th Cir.1994). The Agreement at issue here, however, was assumed by the Debtor early in the case. It has long been the law that if a contract is assumed and then rejected or breached, the resulting damages are treated as an expense of administration. See e.g., Eagle Ins. Co. v. BankVest Capital Corp. (In re BankVest Capital Corp.), 360 F.3d 291, 296 (1st Cir.2004); Adventure Resources Inc. v. Holland (In re Adventure Resources, Inc.), 137 F.3d 786, 798 (4th Cir.1998); Enterprise Energy Corp. v. United States of America (In re Columbia Gas System, Inc.), 50 F.3d 233, 238-239 (3rd Cir.1995); Vision Metals, Inc. v. SMS Demag, Inc. (In re Vision Metals, Inc.), *143 311 B.R. 692, 699 (Bankr.D.Del.2004); In re Sporting Way, Inc., 126 B.R. 110, 112 (Bankr.M.D.Fla.1991); In re Pearson, 90 B.R. 638, 642 (Bankr.D.N.J.1988); In re Monroe Well Service, Inc., 83 B.R. 317, 321 (Bankr.E.D.Pa.1988). This is true both as to pre-petition defaults remaining uncured and damages for post-petition breach. See Adventure Resources, Inc., 137 F.3d at 798; Monroe Well Service, Inc., 83 B.R. at 321. After assumption, the contract is deemed an obligation of the estate. That rationale extends to the consequences of post-assumption rejection and failure to perform. As an obligation of the estate, it is entitled to treatment as expense of administration. See Columbia Gas System, Inc., 50 F.3d at 238-239; see also e.g., In re Resource Technology Corp., 254 B.R. 215, 221 (Bankr.N.D.Ill.2000); Texaco, Inc. v. Louisiana Land and Exploration Co., 136 B.R. 658, 663 (M.D.La.1992); In re Uly-Pak, Inc., 128 B.R. 763, 765 (Bankr.S.D.Ill.1991); Pearson, 90 B.R. at 642. Debtor cites no authority to the contrary. Therefore, Best Western's claim for contractual damages is entitled to be allowed as an administrative expense claim.

B. Liquidated Damages
Best Western claims, as a result of Debtor's continued use of the Best Western trademarks and logos after the Agreement was terminated, that it is entitled to $48,825.00 in liquidated damages pursuant to the formula set forth in paragraph 11[2] of the Agreement. Debtor contends that the liquidated damages clause is unenforceable as a penalty or alternatively, that Best Western improperly applied the formula and miscalculated the date upon which the liquidated damages should commence.
The burden of proof is on the party claiming that a liquidated damages provision is unenforceable as a penalty. See United Merchants and Mfr., Inc. v. Equitable Life Assurance Soc'y of the United States, 674 F.2d 134, 143 (2nd Cir.1982); Balcor Pension Investors v. Wiston XXIV Ltd. P'ship., 170 B.R. 453, 460 (Bankr.D.Kan.1994). Since it is the Debtor urging that the provision is an unenforceable penalty, the Debtor bears the burden of demonstrating that.
The Agreement provides that Arizona law is applicable.[3] This Court must employ the same choice of law principles that a Missouri state court would in determining the applicable law. See Amtech Lighting Services Co. v. Payless Cashways, Inc. (In re Payless Cashways), 203 F.3d 1081, 1084 (8th Cir.2000) citing In re Mastercraft Metals, Inc., 114 B.R. 183, 186 (Bankr.W.D.Mo.1990). In Missouri, a court would honor the contractual choice of law provision if it is not unfair or unreasonable to do so. See Sabatino v. *144 LaSalle Bank, N.A., 96 S.W.3d 113 (Mo.Ct.App.2003); Chase Third Century Leasing Co., Inc. v. Williams, 782 S.W.2d 408, 411 (Mo.Ct.App.1989). Here, the choice of Arizona law is a reasonable one because Best Western is headquartered there. In addition, having a uniform rule applicable to all members of the Best Western network regardless of location would be logical and beneficial. There is no evidence to suggest that applying Arizona law would be unfair or unreasonable.
When determining whether a clause is for liquidated damages or is a penalty, the label placed on the clause by the parties in the contract is not conclusive as to its nature or function. See Aztec Film Productions, Inc. v. Quinn, 116 Ariz. 468, 569 P.2d 1366 (1977) citing Miller Cattle Co. v. Mattice, 38 Ariz. 180, 298 P. 640 (1931). The fact that the Agreement characterizes the provision as one for liquidated damages does not preclude the Court from finding otherwise.
Under Arizona law, it is not necessary for a party relying upon a liquidated damages provision to prove the existence of actual loss or damage. See Mechanical Air Eng'g Co. v. Totem Constr. Co., 166 Ariz. 191, 801 P.2d 426 (1989). A liquidated damages clause will be enforceable if two requirements are met: (1) the amount fixed in the contract must be a reasonable forecast of compensation for the harm caused by the breach; and (2) the harm caused by any breach must be one that is incapable of or very difficult of accurate estimation. Pima Sav. & Loan Ass'n. v. Rampello, 168 Ariz. 297, 812 P.2d 1115, 1118 (1991). The reasonableness of the forecast and the difficulty of estimation must both be viewed as of the date of execution of the contract, not as of the date of the breach. Id. The principal test for determining whether the forecast is a penalty or liquidated damages is whether the payment is for a fixed amount or varies with the nature and extent of the breach. Id.; see also Larson-Hegstrom & Assoc., Inc. v. Jeffries, 145 Ariz. 329, 701 P.2d 587 (1985) citing Miller Cattle Co. v. Mattice, 38 Ariz. 180, 298 P. 640 (1931). An unreasonably large sum for liquidated damages may be unenforceable as a penalty. See Roscoe-Gill v. Newman, 188 Ariz. 483, 937 P.2d 673, 675 (1997) citing Restatement (Second) of Contracts § 356 (1981); Pima Sav. & Loan Ass'n., 812 P.2d at 1117.
In this case, it is clear that the second test regarding the difficulty of estimating the damages is satisfied. Presumably, Best Western's damages are to its reputation in that continued operation of facilities that do not meet Best Western's standards, would cause people not to patronize the facilities of Best Western members and thus reduce their revenues and negatively affect the fees, dues and charges paid to Best Western. Obviously, proving the amount of damages on such a theory would be very difficult, if not impossible.
The contractual formula set forth in paragraph 24 of the Agreement for liquidated damages is 15% of the product of the mean room rate multiplied by the number of rooms the facility has available to rent (in this instance 125 rooms). According to Best Western's calculations, the mean room rate is $84 and the daily liquidated damages amount is $1,575. The Debtor argues that sum is unreasonable and therefore a penalty because it is excessive when viewed either in relation to the amount of contractual fees owed to Best Western on a monthly basis, which is $5,000, or as a percentage of the hotel's average daily sales, which the Debtor calculates to be approximately 72%. Debtor argues that using Best Western's calculations, the amount of the monthly damages *145 is approximately 5.5 times Best Western's loss of fees. This measure is irrelevant, however, because the alleged breach is a violation of a separate provision of the Agreement for unauthorized continued use of trademarks and logos subsequent to termination.
As to Debtor's claim that the amount is unreasonable as a percentage of its average daily room sales, the Court might agree based on Best Western's method of calculation. However, because the Court determines that Debtor's method of calculation is the appropriate one and the daily liquidated damage provision is significantly lower based on that calculation, the Court cannot find that at that level the amount of damages is unreasonable as a matter of law. In addition, the Debtor's calculation is based not on the damages that might have been projected as of the date of the execution of the Agreement, but the actual damages viewed from the date of breach, which is contrary to the law. The concept of basing the damage calculation on a percentage of the Debtor's potential daily revenue is not unreasonable. Presumably, the Debtor's revenues are enhanced by its ability to hold itself out as affiliated with a nationally recognized network such as Best Western. The Debtor has itself taken that position. The actual damage calculation may be high as a percentage of the Debtor's average daily room sales at this time, in part because during the relevant period, the Debtor's occupancy rate may have been abnormally low as a result of having lost the benefit of the national reservation system on April 7, 2005 and as a result of the fire in its kitchen and the unavailability of food services. In this case, the amount of the damages does vary with the nature and extent of the loss as the damages are measured on a daily basis for the period during which Best Western's intellectual property was used without permission after termination.
The Debtor argues that the starting date for the calculation of liquidated damages should be 15 days after April 28, 2005, the date of the hearing on Debtor's objection to the notice of default and the point at which the Court determined that the stay had expired and Best Western had the right to take action to terminate the Agreement. Best Western claims that the starting date for the calculation of damages should be April 23, 2005, fifteen days after the April 7, 2005 effective date of termination, according to its notice. The Court finds that the appropriate starting date is April 23, 2005. Best Western's March 28, 2005 letter was not a notice of default which the Debtor had the opportunity to cure, but rather a notice of termination which Best Western had the right to issue because the Debtor had failed to cure the defaults of which it had been notified in Best Western's letter of February 17, 2005. With regard to the ending date for calculation purposes, the evidence is clear that the Debtor did not terminate its use of all Best Western's signs, logos and trademarks until it removed the last exterior sign from the facility on May 23, 2005.
As the Court observed above, the formula for liquidated damages is based in part upon the calculation of a mean room rate. The Agreement does not define the term "mean." The parties offered several different potential meanings of the term including: a midpoint between two extremes or the average value of a set of numbers. To the extent the Agreement term is ambiguous, that ambiguity should be construed against Best Western as the drafter of the Agreement. See generally United California Bank v. Prudential Ins. Co. of America, 140 Ariz. 238, 681 P.2d 390, 411 (1983).
Debtor's evidence indicated that utilizing either the mid-point or the average approach, *146 the appropriate mean room rate is approximately $49. The Debtor testified as to a list of rates, the middle one of which was $49. The Debtor also testified as to a calculation of the average daily rate based upon the average number of rooms sold and occupancy rates for the months of March and April 2005. The applicable average according to that testimony was $49.02. The fact that these numbers so closely coincide reinforces the reasonableness of the Debtor's interpretation and application of the term.
Best Western claims the appropriate mean room rate is $84, which is based upon rates contained in some guide published by Best Western, based upon information made available by the members. Neither the guide nor the rates themselves were placed into evidence. In addition, the Agreement does not say that the applicable rates are the rates published in the Best Western guide. The Debtor would be the best source of information as to what its actual rates were during the applicable period. The testimony of the Debtor's representative on the issue was uncontradicted. The Court finds that the appropriate daily rate for the liquidated damages is $918.75 per day which is 15% of the product of the mean room rate ($49) multiplied by the 125 available rooms in the Debtor's facility. The resulting calculation of total liquidated damages at the daily rate for the period from April 23, 2005 through and including May 23, 2005 is $28,481.25.

C. Attorneys' Fees
Best Western claims it is entitled to be reimbursed for attorneys' fees incurred in this proceeding in the amount of $32,221.75. Because the Debtor assumed the Agreement, it is bound by all of its provisions. One of them, paragraph 36, states that if a member breaches any obligation to Best Western, it is liable to Best Western for attorneys' fees, costs and expenses incurred by Best Western "in connection with the breach." Attorneys' fees incurred by a non-debtor party to an agreement following default may be recouped, upon assumption, if expended as a result of the default. In re Freckleton, 2001 WL 34050700, *2 (Bankr.D.Vt.2001). These claims must relate to the contractual provision that allows payment. Freckleton, 2001 WL 34050700 at *3; In re Health Science Products, Inc., 191 B.R. 895, 910 (Bankr.N.D.Ala.1995). The fees sought must also be reasonable. Freckleton, 2001 WL 34050700 at *3; Health Science Products, 191 B.R. at 910 n. 20.
Best Western submitted an affidavit of its counsel along with an itemized statement of fees and expenses. The Court must determine whether these fees were incurred "in connection with the breach" and whether they are reasonable. After reviewing the statement, the Court finds that the vast majority of the entries meet both requirements. The Court, however, finds that certain of the tasks performed relate generally to representation of Best Western in the Chapter 11 proceeding and are not related to any alleged breach of the Agreement. In that category are entries for time spent providing status reports to the client on the Chapter 11 proceeding and review of unspecified documents. For those reasons, the Court disallows the following time entries:


----------------------------------------------------------
                       Time
Date                   Entry       Reason
----------------------------------------------------------
November 18, 2004         .2       Unspecified
----------------------------------------------------------
December 14, 2004         .3       Status report to client
----------------------------------------------------------
December 17, 2004         .9       Miscellaneous matters
----------------------------------------------------------
January 6, 2005           .5       Miscellaneous matters
----------------------------------------------------------
January 18, 2005          .3       Status report to client
----------------------------------------------------------
March 4, 2005             .2       Review of order
                                    relating to claim
                                    deadline
----------------------------------------------------------
*147
March 26, 2005            .3       Status report to client
----------------------------------------------------------
April 8, 2005            1.0       Miscellaneous matters
----------------------------------------------------------
April 29, 2005            .4       Status report to client
----------------------------------------------------------

The total number of hours reflected in these entries is 3.1. At Mr. Helms' hourly rate of $225, the value of these time entries is $697.50.
A substantial portion of several additional entries is for time traveling between Kansas City and Phoenix for the numerous hearings conducted in the case. There is no indication that any work was being performed during these periods. The Court finds that it is appropriate to discount that time and allow it at half counsel's normal hourly rate. See Bachman v. Laughlin (In re McKeeman), 236 B.R. 667, 672 (8th Cir. BAP 1999) (unreasonable to charge full hourly rate for travel time) citing In re Anderson Grain Corp., 222 B.R. 528, 532 (Bankr.N.D.Tex.1998) (same); In re Bennett Funding Group, Inc., 213 B.R. 234, 251 (Bankr.N.D.N.Y.1997) (allowing one-half normal rate unless work performed during the travel). Accordingly, the following entries (or portions thereof) will be reduced by one-half of the travel time included.[4]


----------------------------------------
               Original          Reduced
               Time              Time
Date           Entry             Entry
----------------------------------------
11/29          5.0               3.5
----------------------------------------
11/30          8.0               6.5
----------------------------------------
1/23           3.0               1.5
----------------------------------------
1/24           8.0               6.5
----------------------------------------
4/26           8.0               6.5
----------------------------------------
4/27           11.0              9.5
----------------------------------------

The time reduction amounts to nine hours, which at Mr. Helms' $225 hourly rate totals $2,025.00. The total fee reduction is $2,722.50. The Court will therefore allow attorneys' fees and costs to Best Western in the amount of $29,498.85.

III. CONCLUSION AND ORDER
For all of the reasons stated above, the Court will grant Best Western's Request for Allowance and Payment of Administrative Expenses in part and overrule it in part. Best Western shall have an administrative expense claim against the Debtor's estate for (1) contractual damages of $51,094.16 (2) liquidated damages of $28,481.25, and (3) attorneys' fees and costs of $29,498.85. This Memorandum Opinion constitutes the Court's Findings of Fact and Conclusions of Law as required by Rule 9014(c) and Rule 7052 of the Federal Rules of Bankruptcy Procedure. A separate order will be entered as required by Rule 9021.
NOTES
[1]  See Movant's Exhibit A; Affidavit of Cheryl Pollack, p. 2, ¶ 3.
[2]  Paragraph 11 of the Agreement provides: for each day during which any Best Western Symbol or any name, symbol or devise described in paragraph 23 are used in connection with the Hotel, after fifteen (15) days following termination of this License Agreement, Best Western may elect to claim from Applicant daily damages in an amount equal to fifteen percent (15%) of the mean of the Hotel's room rates per room per day multiplied by the total number of rooms. This amount is payable by Applicant whether or not Applicant continues to exercise control over the operations of the Hotel. It is understood and agreed that said amount is fixed as liquidated damages because of the difficulty of ascertaining the exact amount of damages that may be sustained because of such use. It is further understood and agreed that said amount fixed as liquidated damages is a reasonable amount, considering the damages that Best Western will sustain in the event of such use.
[3]  See ¶ s 37 and 38 of the Agreement.
[4]  Some of the entries containing descriptions which include travel, do not separately identify the travel time. In examining other entries, the Court has determined that the travel time between the two cities is approximately three hours and has operated on that assumption in making the reductions reflected in the table in the text.
