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[S2M-2931.3 (Amendment)] Decision Time — 9 Jun 2005 at 17:30
This looks like the vote on S2M-2931.3
The description in the bulletin on 2005-06-09 is:
*S2M-2931.3 Mr Andy Kerr: Health Issues with Specific Reference to the Kerr Report—As an amendment to motion (S2M-2931) in the name of David McLetchie, leave out from "welcomes" to end and insert "commends the Kerr report, Building a Health Service Fit for the Future, and its emphasis on sustainable and safe local services and preventative care for the most vulnerable, which supports the Scottish Executive's goal of delivering care that is as local as possible and as specialised as necessary; welcomes the interim report of the NHS 24 review team and the public commitment of NHS 24 to implement the key recommendations as quickly as possible for the benefit of the people using the service, and is encouraged by the positive progress on waiting times and waiting lists." Supported by: Rhona Brankin* Motions and amendments which have attracted additional support
You can search for this motion (S2M-2931.3) on TheyWorkForYou
Text Introducing Division:
The first question is, that amendment S2M-2931.3, in the name of Andy Kerr, which seeks to amend motion S2M-2931, in the name of David McLetchie, on health issues, with specific reference to the Kerr report, be agreed to. Are we agreed?
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MPs for which their vote in this division differed from the majority vote of their party are marked in red. Also shows which MPs were ministers at the time of this vote. You can also see every eligible MP including those who did not vote in this division.
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Person Info
William Gutterson: Birth: 31 OCT 1697 in Andover, Essex Co., MA. Death: 2 SEP 1742 in Methuen, Essex Co., MA
Joseph Gutterson: Birth: ABT 1709 in Andover, Essex Co., MA.
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LANSING, Mich. (WJMN/Press Release) – Ameer is a photographer who loves the Avengers movies. Le'Asia wants to demonstrate how to do the perfect cartwheel. Jules has a passion for colorful makeup.
As the state celebrates Adoption Month as proclaimed by Gov. Gretchen Whitmer during November, the three Michigan teens are among those looking for forever families. Ameer, Le'Asia and Jules, in need of adoptive homes, are featured on the "20 in 2020: Meet the Youth" website, which includes in-depth self-directed stories from the teens using video, photos and their own words.
"Michigan is committed to finding loving adoptive families for youth in foster care as soon as possible," Gov. Whitmer said. "I hope that Michiganders will consider adoption."
MDHHS partnered with the Michigan Adoption Resource Exchange, which created the 20 in 2020 project that launched in March. The goal is to produce videos for 20 teens who are awaiting adoptive families by the end of the year.
"Every child deserves a forever family," said JooYeun Chang, executive director of the Children's Services Agency at the Michigan Department of Health and Human Services (MDHHS). "We have a special request for families who are willing to step forward and provide adoptive homes for teens in our foster care system. These incredible young people have talents, hopes and dreams. We invite you to go online and learn more about them."
Through 20 in 2020, youth are empowered to be decision-makers and creators, choosing what they want to share as a part of the family search process. They work one-on-one with Michigan Adoption Resource Exchange staff, from initial brainstorm to final review, to create videos, photos, podcasts, narratives, artwork or anything else they feel best represents them.
Sixteen of the 20 profiles have been created. Five of the children already have been matched with adoptive families or have pending adoptive placements.
Other partners in the project are AdoptUSKids and the Northwest Adoption Exchange. It's part of the U.S. Department of Health and Human Services Children's Bureau Family Finding and Engagement Initiative.
Approximately 12,100 children are in foster care in Michigan, including more than 300 children who still need a forever family through adoption. That's down from 13,710 in 2018 as MDHHS has focused on keeping families together, reunifying children safely with their families, and finding adoptive homes more quickly when safe reunification is not possible. More than 1,700 children were adopted from Michigan's child welfare system in fiscal year 2020.
Courts statewide, in partnership with MDHHS, are holding special Adoption Month ceremonies in November at which children officially become permanent members of their new forever families. Included is a Nov. 24 Michigan Supreme Court event that will be held virtually this year. A list of Adoption Day events and profiles of adoptive families can be found on the Michigan Adoption Resource Exchange website.
Anyone interested in adopting from foster care can contact the Michigan Adoption Resource Exchange at 800-589-6273, or see biographies and photos of children available for adoption at www.mare.org.
Latest Stories
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Home of the Scrapbook Campus › Forums › Showroom › What are you working on (in July)? › Reply To: What are you working on (in July)?
July 9, 2018 at 9:31 am #20080
- 1097
- Superfan
Michele, I guess this happens to all of us… There are times I know exactly what I want to make but then other times… well you know 🙂


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The Clark County Coroner's office wants to appeal a judge's order to provide autopsies to the Las Vegas Review-Journal despite spending more than $75,000 in taxpayer money.
After a failed attempt to release an improved disease investigation platform this fall, Nevada says the state will have to wait until summer 2021.
Las Vegas police union officials suggested that the Metropolitan Police Department has been overeager in asking employees to stay home. The sheriff refuted.
Brig Lawson, former executive with the Las Vegas Convention and Visitors Authority, is the last of four officials to agree to pay fines for personal use of agency gift cards.
Criminals are regularly released without making full restitution to their victims. Flawed policies and offenders who clearly don't have the money to pay are to blame.
Republicans have alleged widespread voter fraud because of Nevada's mail-in ballots. A review of facts found few irregularities that could have swayed the results.
The number stretches back to June. Until now, the visitor data ran through only mid-August. At that time at least 530 visitors had tested positive for COVID-19.
Controversy has swirled around the question all year. Lacking a national coronavirus death definition, state officials created their own.
District Judge Judge Jim Crockett blasted the coroner's office for failing to release the autopsies and accused the agency of "heel-dragging."
Police are conducting a criminal investigation into allegations that a Las Vegas assemblyman misused campaign funds and failed to live in his district, the Review-Journal has learned.
After months of lockdowns, social isolation and mask-wearing, Nevada health officials are concerned about residents ignoring coronavirus precautions.
Johns Hopkins University published an alarming COVID-19 positivity rate that puts Nevada well above the national average. It's also incorrect, state officials say.
Phillip Merrill, a longtime friend of suspected boogaloo member Stephen Parshall, pleaded guilty in July to sexually assaulting the alleged extremist's stepdaughter.
The listing follows the sale of more than $5 million worth of other properties. Adolfo Orozco faces involuntary manslaughter charges in connection with the downtown fire.
Dana White's lawyers had filed court papers seeking to toss out the suit, calling it a "smear" effort and new attempt to extort money from White through the civil court system.


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Numerical evaluation of loop integrals
We present a new method for the numerical evaluation of arbitrary loop integrals in dimensional regularization. We first derive Mellin-Barnes integral representations and apply an algorithmic technique, based on the Cauchy theorem, to extract the divergent parts in the limit. We then perform an -expansion and evaluate the integral coefficients of the expansion numerically. The method yields stable results in physical kinematic regions avoiding intricate analytic continuations. It can also be applied to evaluate both scalar and tensor integrals without employing reduction methods. We demonstrate our method with specific examples of infrared divergent integrals with many kinematic scales, such as two-loop and three-loop box integrals and tensor integrals of rank six for the one-loop hexagon topology.
I Introduction
Perturbative methods are indispensable in order to establish consistent theories of particle interactions, and to predict quantitatively their experimental manifestations. The anticipation of new phenomena in modern experiments and theoretical extensions of the Standard Model, requires cross-sections for complicated processes. This has driven a remarkable progress in the development of new computational methods. At the one-loop level, the calculation of cross-sections with five external legs is gradually becoming a routine activity (e.g. nlo_example ). At two-loops, there have been recent successful computations of amplitudes with four external legs and up to three kinematic scales (e.g. nnlo_example ). At three-loops and beyond, amplitudes with up to one parametric variable have also been computed (e.g. bnnlo_example ).
Many new processes with higher final-state multiplicity, number of loops, and kinematic scales, have been identified to be important at the TeV energy frontier. The aim of our paper is to provide a new method which can be used to compute loop amplitudes for such, more complicated, processes.
Loop integrations are cumbersome due to the presence of infrared singularities. Loop integrals with many kinematic scales have, in addition, a complicated analytic structure with respect to their kinematic parameters. The tensor structure in gauge theories is also an issue, since it proliferates the number of terms. A general method for computing arbitrary loop integrals should extract their infrared and ultraviolet singularities, and treat simply kinematic discontinuities and threshold singularities. For practical applications, it should also be able to handle tensor integrals efficiently. There is no method which addresses satisfactorily all these issues; known techniques can compute a limited number of amplitudes where some simplifications occur in special cases.
Following a traditional approach to calculate loop amplitudes, one reduces the number of terms to a few master integrals and computes the latter analytically. One-loop integrals can be reduced using the classical method of Passarino and Veltman pvreduction . For generic multi-loop computations one can derive reduction identities from integration by parts ibp and the invariance of scalar integrals under Lorentz transformations li . There is a large variety of approaches for evaluating the master integrals analytically. For example, one can compute integrals with a simple singularity structure and a small number of kinematic scales by integrating directly their Feynman parameter representation. For more complicated cases one can use advanced techniques, such as the method of differential equations kotikov ; lance ; li .
This approach can fail, however, if we apply it to complicated processes. The reduction algebra is hard, and the expressions of amplitudes in terms of master integrals may have spurious singularities which hamper their numerical evaluation. The extraction of poles in the master integrals, the evaluation of the coefficients of the -expansion in terms of known analytic functions, and the analytic continuation of the latter to physically interesting kinematic regions are also involved. It is, thus, very well motivated to improve or replace the "traditional scheme" and to develop new automated methods.
One-loop amplitudes can be entirely determined in four dimensions in terms of basic functions such as logarithms and polylogarithms that appear in the one-loop scalar box. New methods introduce sophisticated algorithms for the reduction to the basic functions; by either numerical or analytical techniques, they control the appearance of spurious singular terms and minimize the size of the intermediate expressions glovergiele ; denner ; heinrich_oneloop ; Binoth:2001vm ; weinzierl ; pittau . Recently, cross-sections for processes at NLO were computed denner_result using such techniques. A method, which is inspired by techniques for real radiation at NLO, renders one-loop graphs numerically integrable nagy with the introduction of universal subtraction terms. A different approach uses unitarity, dualities, and analyticity properties for the determination of one-loop amplitudes bdk .
Beyond one-loop, there has been a significant progress in the automation of the reduction methods to master integrals laporta ; air ; grobner . The infrared structure of multi-loop integrals is substantially more complex than at one-loop; the development of methods for their numerical evaluation is more difficult. Nevertheless, there is significant progress in this direction passarino . A powerful numerical method for multi-loop calculations is the method of sector decomposition sector_loop ; Binoth:2003ak ; Hepp:1966eg ; it simplifies recursively the singularities of Feynman parameterizations and allows a straightforward expansion in . The method of sector decomposition has been very succesfully employed in the computation of several multi-leg integrals at one, two and three loops heinrich_oneloop ; sector_loop ; Binoth:2003ak . This method has been introduced, recently, for the purely numerical evaluation of multi-loop amplitudes muon . However, it is perplexing how to apply it for loop-integrals in non-Euclidean regions.
In 1999, Smirnov smirnov and, soon later, Tausk tausk , introduced a new method for the evaluation of loop integrals. In their pioneering papers, Smirnov and Tausk smirnov ; tausk computed analytically the first infrared divergent double box integrals. In veretin ; tausk1 new two-loop integrals for massless processes were computed. In smirnovB the method was applied to double-box integrals with one additional mass-scale. The method was spectacularly applied in the computation of three-loop amplitudes smirnovC ; smirnovD in supersymmetric Yang-Mills theory, shedding light to novel cross-order perturbative relations of the theory smirnovD ; msym .
The Smirnov-Tausk method is based on a few simple ideas. Starting from the Feynman parameterization of a loop integral we can derive a new representation in terms of a multiple complex contour integral. Such, Mellin-Barnes (MB), parameterizations have enabled compicated loop calculations by using powerful methods for complex integration usyukina . Smirnov and Tausk exploited a novel property of these representations. Infrared divergences localize on simple poles inside the complex integration volume. We can isolate the divergent pieces of the integral at , by using the Cauchy theorem. After subtracting the divergent residues, we can perform a Taylor expansion in , and sum up the remaining infinite series of residues. Finally, we can work to derive analytic expressions for the infinite sums in the coefficients of the expansion in terms of logarithms, generalized polylogarithms remiddi_vermaseren , and more complicated functions.
The Smirnov-Tausk method is very powerful; however, it is laborious and intricate. The isolation of the divergent residues in multiple Mellin-Barnes integrals is convoluted. In addition, it is difficult to identify infinite sums in terms of polylogarithm functions with known analyticity properties. As a consequence, the analytic continuation in physical kinematic regions is also involved. Due to these complications, the method has been applied to a few master integrals with a small number of kinematic scales. In this paper, we generalize the method to a broader spectrum of applications.
As a first task, we automate the procedure for the isolation of the divergent residues at . Smirnov smirnov and Tausk tausk use different techniques for finding these residues. The approach of Smirnov is very intuitive, but daedal. We have found that the technique described by Tausk in Ref. tausk resembles closely to a programmable algorithm. We have used it as a guide and we have written computer programs which subtract the poles in arbitrary Mellin-Barnes integrals.
In our method, we avoid entirely the painstaking tasks of finding analytic expressions for infinite sums in terms of polylogarithms, and performing the analytic continuation in the arguments of polylogarithms. Mellin-Barnes representations are valid in kinematic regions where loop integrals may be complex-valued. We have found that, in a broad spectrum of applications, it is simple to calculate the representations numerically. The only analytic continuation that is ever required is that of logarithms with a single kinematic scale as an argument.
An important goal of our method is to calculate loop amplitudes in realistic gauge theories. We have found that tensor integrals and, furthermore, diagrams which belong to the same topology can be calculated collectively. As we will show, the integrand of a representation for a scalar integral will be a product of Gamma functions and powers of kinematic invariants, while that of a generic tensor will be the same integrand as in the scalar integral multiplied by a polynomial in the integration variables. The evaluation of polynomials is fast in a numerical program; the computational cost for evaluating tensor integrals or loop diagrams is not significantly larger than evaluating scalar integrals. The only practical issue that we need to address, is the book-keeping of the various terms that contribute to the polynomial; we present an efficient solution of this problem here.
In this paper, we apply our method to a number of examples. We, first, test our method in scalar and tensor integrals of the one-loop massless hexagon topology. The purpose of this computation is to introduce our method for tensor integrals and to demonstrate that we can tackle problems which are relevant in computations of physical amplitudes. We present here results for tensors through rank six, in both the Euclidean and the physical region for processes. The numerical programs that we have constructed are suitable for the evaluation of the QCD amplitudes in four-jet production at hadron colliders.
In the second set of examples, we compute scalar two and three-loop integrals which are known analytically in all kinematic regions: the massless planar smirnov and cross tausk double-box, the massless double-box with one off-shell leg thomas_int ; harmpol1 ; harmpol2 ; harmpol_ac ; smirnovB , and the massless planar triple-box smirnovC . They serve to cross-check our algorithms and to demonstrate that we can easily reproduce state-of-the-art computations. Our numerical results are in excellent agreement with the analytic expressions. We also present a number of new results that would require significant efforts for their computation with traditional approaches. We present, for the first time, double-box integrals with up to four kinematic parameters, and triple-box integrals with up to three kinematic parameters computed in all physical regions.
In Section II, we explain our technique for deriving Mellin-Barnes representations for loop integrals. In Section III, we describe our routines for performing an -expansion of Mellin-Barnes representations for scalar integrals. We extend these results to loop integrals with tensor numerators in Section IV and, in Section V, we present our methods for the numerical integration. Section VI is devoted to present our results for several integrals at the one, two and three loop level. Finally we present our conclusions.
Ii Mellin Barnes representations
We start with a brief discussion on the derivation of Mellin-Barnes representations for loop integrals from their Feynman parameterization. The construction of parameterizations is not unique, and various representations of the same integral may have quite different features. For example, they could have a different integral dimensionality, or they could be better or worse suited for numerical evaluation. A valid parameterization, however, can always be found. A pedagogical introduction to the topic of Mellin-Barnes representations is presented in Ref. smirnov_book .
As a concrete example we consider the one-loop box integral with two adjacent external legs off-shell and massless propagators,
and , , , , . The powers of the propagators and the dimension are kept arbitrary and we will derive a Mellin-Barnes parameterization of the box-integral in this general case. In this section, we will seek values for where the representation is well defined and the integral is finite. In the next Section, we will explain the technique for the analytic continuation to the values of the parameters, such as , where the integral develops divergences.
Our starting point is the Feynman parameterization of the box integral:
with . The main tool for obtaining the Mellin-Barnes representation of the above integral is the formula:
where the following conditions are satisfied:
are complex numbers with ,
the contour of integration is a straight line parallel to the imaginary axis separating the poles of and , i.e. .
We can use the identity of Eq. 3 to simplify the entangled denominator of Eq. 2, with the cost of introducing new integrations in the complex plane. Before we do so, we would like to point out an attractive feature of Eq. 3. The above two conditions guarantee that the integrand in the right hand side of Eq. 3 vanishes at infinity. We can then close the contour of integration at infinity and calculate the integral using the Cauchy theorem. If we close the contour to the side of the positive real-axis we obtain
This is the Taylor expansion of the left hand side of Eq. 3 in the region . Equivalently, by closing the contour of integration to the side of the negative real axis, we obtain a Taylor expansion in the complementary region . This is a particularly useful property: representations of Feynman integrals which are derived by using the Mellin-Barnes decomposition are valid in all kinematic regions. In addition, can be complex; therefore, we can account for the infinitesimal imaginary part that is assigned to invariant masses and kinematic parameters. Eq. 3 may be recursively applied to denominators with more than two terms, yielding:
We are now ready to apply the decomposition of Eq. 5 to the denominator of Eq. 2, introducing three contour integrations:
where the contours must satisfy condition 2 above, i.e. they must be chosen in such a way that the arguments of the Gamma functions are all positive. It is now straightforward to integrate out the Feynman parameters, using
To exchange the order of integrations, we must assume that the exponents of the Feynman parameters satisfy
With these conditions the Gamma functions on the right hand side of Eq.7 have arguments with positive real part as well. The final result is:
and we have adopted the shorthand notation
In general, Mellin-Barnes representations for a loop-integral are well defined for number of dimensions and powers of propagators which render the real parts of the arguments in all Gamma functions positive. For example, it is impossible to satisfy positivity in all Gamma function arguments for . This is in accordance with the expectation that the scalar box integral is infrared divergent in four dimensions. But, for example, the representation of Eq. 9 is well defined if we choose the contour and the parameters . In the next section we will detail the analytic continuation from values of parameters for which the Mellin-Barnes representation is well-defined to values for which the integral develops divergences. That is, a procedure which makes the divergences appear explicitly in the form of poles in the parameter space.
From Eq. 9 we observe that it is simple to implement the analytic continuation of kinematic parameters in a numerical evaluation, since they always appear in a factorized form, e.g.
We need the analytic continuation of simple logarithms; these are evaluated trivially in all kinematic regions:
It is clear from the above example that we can derive a Mellin-Barnes representation for any integral starting from its Feynman parameterization. The form of the parameterization depends on the choice of Feynman parameters, the order in which the terms appear in the right hand side of Eq. 5, and the implementation of the delta function constraint for the Feynman parameters. This arbitrariness is more pronounced beyond one-loop.
In Section IV we will derive representations for tensor one-loop integrals which simplify their numerical evaluation. We aim to find similarly simple representations for multi-loop tensor integrals as well. For this purpose, it is convenient to use the one-loop representations as building blocks, by employing a re-insertion technique tausk1 ; usyukina . We explain this technique in our second example.
We now derive a Mellin-Barnes representation for the massless planar double-box integral with two adjacent legs off-shell. The integral is defined:
and , , , , . For simplicity, we have set the powers of propagators to one. We perform, first, the -loop integral and derive the representation for this integral reading it off from the one-loop result of Eq. 9,
the off-shell legs of the -loop box correspond to the propagators while the invariant mass gives rise to a new propagator for the second integration:
The propagators form a new one-loop box with two adjacent legs off-shell, and we can read off, once again, the representation for this second integration using Eq. 9. The final result for the two-loop box with two adjacent legs off-shell is:
We have produced a representation for a two-loop integral by embedding representations for one-loop integrals into other representations. It is obvious that we can use the re-insertion method for writing representations for arbitrary multi-loop integrals.
Representations for integrals with given kinematic scales can be used to derive, easily, results for simpler integrals where some of the scales are taken to zero. Let us, for illustration, derive the representation for the double-box with one-leg off-shell and the on-shell double-box. We must first take the limit in Eq. 14. The term is vanishing in this limit unless at the same time. If we use the Cauchy theorem, we find that the wanted limit is given by taking the residue of the integrand at . Therefore, the double-box with one-leg off-shell is:
Similarly, the on-shell double-box is given by:
Note that, in order to have a finite limit for a vanishing kinematic scale, the Mellin-Barnes representation should have terms in the integrand which behave as . These emerge naturally in non-trivial representations due to Eq. 3.
Iii Analytic continuation
In the previous Section we have noted that the Mellin-Barnes representation of a loop integral is valid if appropriate poles of the Gamma functions lay separated on the right and left of the integration contours. This condition guarantees the equivalence between the Mellin-Barnes integral and the original denominator in the loop integral. It often occurs that this condition can not be satisfied for values of the dimension parameter and the powers of the propagators which are relevant for a realistic application. Let us consider, as an example, the one-loop box representation of Eq. 9 in the case of powers of propagators set to unity and the dimension in ,
This representation can only be valid for values of different from zero. For example, if we choose the contour we find that should be in the interval .
We can use the Cauchy theorem to obtain a representation in terms of a sum of contour integrals, which is valid at and the poles appear explicitly. The key point is that if the value of is chosen outside the allowed region, for example , some of the poles of the Gamma functions will be on the wrong side of the contours (see Fig. 1). To recover the original result, we must compare the position of the poles relative to the contours for values of inside the allowed region and the point outside. Depending whether the poles crossed the contours from left or right, we should correct the representation by adding or subtracting the residue of the integrand on those poles. This simple observation can be easily cast into general purpose algorithms which extract the poles in of arbitrary Mellin-Barnes representations.
Let us choose a value of inside the allowed region and observe the position of the poles with respect to the contours for some -dependent Gamma functions in our example. The poles of and , at and , are situated to the right of the and contours, respectively, for all non-negative integers . Let us now take . We observe that the first poles, for , moved to new positions, and , which are on the left side of the contours. The poles for remain on the right side. Therefore, the representation is not valid at , since the contours separate poles which originate from the same Gamma function. To obtain a valid expression for the integral at , we need to use the Cauchy theorem to isolate the crossing poles ().
Smirnov, in Ref. smirnov_book , provides a number of pedagogical examples where this is done. The approach of Smirnov is to deform the multiple contours, using the Cauchy theorem, so that the "offending" poles never cross the contour of integration. Tausk, in Ref. tausk , describes in detail this task as well, using as an explicit example the Mellin-Barnes representation for the two-loop cross box integral. The technique of Tausk is different from the one of Smirnov; the main idea is to account for the poles that end up on the "wrong" side of the contour in a progressive way, as they cross the contours when changing continuously the value of . In this paper, we have followed the approach of Tausk. We would like to recommend that the interested reader studies the pedagogical example of Ref. tausk . Here we discuss our implementation of the algorithm which is depicted in the flow diagram of Figure 2.
Consider the Mellin-Barnes representation of a loop integral. The integrand of depends initially on Gamma functions. The application of the analytic continuation algorithm requires the iterative evaluation of residues. These may also give rise to Psi functions, which have poles in the same positions the Gamma functions do, and are treated identically for the purposes of analytic continuation. The contours of the representation are straight lines parallel to the imaginary axis, and are chosen so that the real parts of the arguments of all Gamma functions are positive. The same condition determines an allowed region for , which we will assume that includes the point . For the sake of clarity, we will consider the case , but the method proceeds analogously when .
The first step is to find the maximum value , with , for which the representation holds. At this value, one or some of the positions of the poles run into a contour. If the set of these poles, , is empty, i.e. no poles cross the contours when moving from to 0, then the integral can be safely expanded around . So, let us focus on the more interesting case in which is non empty. Here we have to distinguish between and .
If , at least one pole lies on the contours for the terminating value of . Then, the representation has an unregulated singularity, and it is not feasible to perform an expansion and evaluate the contour integrals numerically. To remedy the situation we must change the contour of integration; we use the Cauchy theorem and displace one or more of the contours parallel to the imaginary axis. Then, we attempt afresh the analytic continuation; the poles should now hit the contour at . If this does not happen, due to an unfortunate choice of the contour we displaced, we simply iterate the procedure until we succeed.
When , we have to determine the residues in the poles that cross the contour. If there is only one pole in the situation is simple. Let us look back at the specific example of the one-loop box. By applying the previous steps of the algorithm, we find that the first pole to cross a contour is at . We then choose one of the multiple integrals, e.g. the integral, and subtract the residue of the integrand on this pole, at , to the representation:
The residue term in the above expression is a new multiple Mellin-Barnes integral with one integration variable less than in . In our example, the pole crosses the contour from the right side and we subtract its residue from the original representation. If a pole crosses the contour from the left, we should add its residue. At this point, we have achieved the analytic continuation in from to . We now repeat the algorithm setting as initial value , and perform the analytic continuation to the next value of where a new crossing occurs. We proceed iteratively, until we find all poles which cross the contours of integration in the terms of Eq. 18 and its descendants.
The algorithm is more complicated if many poles cross simultaneously the contour at . In our one-loop box example, if we had chosen a symmetric contour then we would have two poles crossing at the same time: . One way to deal with this situation is to displace the contours in such a way that the degeneracy is removed, i.e. introduce asymmetric contours from the beginning. Another alternative is to take combined residues in all the poles. To correctly account for all the terms in this case, we can imagine that the contours are only slightly displaced from their current position. Then the poles will hit them one at the time and we can take residues successively. Finally, we can restore the original position of the contours without affecting the position of the poles.
There is, however, a caveat in taking multiple residues. Let us consider the case of the one-loop box, where the poles and hit the contour at the same time. We take residues in sequence, considering first . We make the "unfortunate" choice to take the residue on the integration variable, and not on . The analytic continuation yields:
Then we consider the crossing of the pole on each of the two terms on the right hand side of the above expression. We find no problems in taking the residue of the first term. However, the position of the pole for the second term is now written:
The pole is now independent of and gets nailed to the contour. This situation is similar to the one we encountered for , and we can deal with it in exactly the same way by displacing the contours.
We apply the algorithm recursively, to all the terms which are generated by adding the residues of crossing poles to the original parameterization. At the end of this procedure, we obtain a sum of contour integrals plus some terms that do not contain any remaining integral due to taking residues on all integration variables. The method assures that the sum of all these contributions equals the original loop integral when . So we can expand in a power series of .
In this Section, we have presented the algorithm assuming that only the dimension parameter is required to regulate the Mellin-Barnes parameterization. However, the method proceeds identically for integrals where multiple analytic regulators are required. This could be the case in non-covariant gauges or in integrals with irreducible numerators tausk1 or additional positive powers of propagators veretin .
The actual implementation of the algorithm in a set of routines in both MAPLE and MATHEMATICA is very efficient, and allows to perform the analytic continuation in the regulator parameters in a few seconds, for all the Mellin-Barnes representations that we have studied in this paper.
Iv Mellin Barnes representations for tensor integrals
Mellin-Barnes representations, with the exception of the massless two-loop diagonal box topology veretin , have been traditionally employed for the evaluation of master integrals. In this Section, we introduce an efficient generalization of the method to tensor integrals; our method does not require any reduction techniques to master integrals. Common reduction methods produce extremely large algebraic expressions which may also have spurious singular denominators. Here we derive efficient representations for one-loop tensor integrals, which are amenable to numerical integration. We can derive analogous representations for multi-loop integrals by using the re-insertion method of Section II.
We consider a generic one-loop tensor integral of rank with external legs:
and are external incoming momenta. We first introduce Feynman parameters for the denominator. We obtain,
Now we perform the usual shift in the loop momentum, , and we obtain,
where, we denote
We find the standard Feynman representation for the generic tensor one-loop -point function, by integrating the loop momentum:
where for odd, and for even. We observe that the sum, in the second line of the equation, is a polynomial in the Feynman parameters, with tensor coefficients:
For the scalar integral with unit powers of propagators, this sum is .
We will use Eq. 26 to derive Mellin-Barnes representations for the tensor loop-integral. We would like to exploit the fact that the Feynman representation of the scalar and tensor integrals have the same denominator:
In Section II, we have seen that we can obtain a Mellin-Barnes representation for a loop integral by decomposing the denominator of the Feynman parameterization with Eq. 5. We should anticipate that the Mellin-Barnes representations of tensors and scalars are, therefore, very similar. We can verify this intuition, if we consider one generic term in the polynomial of Eq. 27, and follow the steps of Section II. Decomposing the denominator with Eq. 5, we obtain
where is the number of kinematic scales in the integral. The factors are universal for all terms in Eq. 27, and originate from the dependence of on the Feynman parameters. We now proceed to integrate out the Feynman parameters using Eq. 7. This integration yields Gamma functions
Rewriting all Gamma functions as above in Eq. 27, we obtain, for the tensor integral, the Gamma functions of the scalar integral multiplied by simple factors with Mellin-Barnes integration variables.
The result for the one-loop tensor integrals depends on the topology and the rank of the tensors in a rather complicated manner. However, we find that the representation is of the following form
The first line of the above representation contains all the terms which already appear in the analogous representation of the scalar integral with unit powers of propagators. The function is a polynomial in the Mellin-Barnes variables, with tensor coefficients in terms of the external momenta. This is very important for our strategy to evaluate tensor integrals and Feynman diagrams. The polynomial in the numerator is smooth, and it does not affect the analytic continuation in . We can therefore perform the continuation collectively for all tensor integrals and diagrams of the same topology.
We organize the evaluation of tensor integrals and diagrams of a topology as follows. We first find the Mellin-Barnes parameterization of the scalar integral of the topology, and multiply the integrand with a template function which we assume depends on all Mellin-Barnes variables. Then, we apply the analytic continuation algorithm of Section III keeping general. We expand in , and create numerical programs for the evaluation of the expansion for any smooth function . This part of the evaluation needs to be performed only once for a given topology. Then we must identify the polynomial for the evaluation of the specific tensors or diagrams that we are interested in. We have written FORM form and MATHEMATICA programs which perform the steps that we described in this Section, and derive the explicit functional form of the polynomials in Eq. 30. We then write numerical routines for their evaluation, and link them to the general routines for the expansion of the integrals of the topology.
This approach is efficient for the application of our technique to tensor integrals. It turns out, however, that the expressions for the polynomials are quite long for high rank tensors. We have observed that we can reduce the size of the expressions substantially with a simple modification. The terms in Eq. 26 are lengthy, and when we integrate out the Feynman parameters they give rise to large expressions in of Eq. 30. We rewrite Eq. 26 in a different way,
Then, we introduce a Mellin-Barnes decomposition for each denominator:
and integrate out the Feynman parameters. At the end of this procedure, we obtain a sum of Mellin-Barnes integrals for a tensor of rank . All integrals correspond to the Mellin-Barnes representation of the scalar integral with shifted dimension . Each of them has a different polynomial in the numerator. However, these are substantially shorter than the polynomial in Eq. 30.
It is worth to note that, with our method, we never introduce spurious singularities. The evaluation of the higher rank tensors and diagrams in gauge theories is very efficient, since we never require the manipulation of lengthy expressions. The polynomials can be large, however, they require a minimum amount of handling before generating numerical codes for their evaluation.
Here we have presented the derivation of efficient representations for one-loop integrals. The method may be applied to multi-loop integrals using the one-loop tensor integral representations as building blocks. The analogous functions of in multi-loop tensors contain Gamma functions in the denominator. However, they are smooth and maintain the properties which are required for a collective evaluation of all tensors of a topology. Explicit applications with multi-loop tensor integrals will be examined in future work.
V Numerical evaluation of Mellin-Barnes representations
After analytic continuation, the contour integrals can be safely expanded in power series of . The coefficients of the expansion are also contour integrals which, in general, contain in their integrand Gamma functions, Gamma function derivatives, simple logarithms, and powers of the kinematic parameters. In earlier works, these integrals were computed analytically. The standard approach is to evaluate them by means of the residue theorem, i.e. summing an infinite number of residues and expressing the power series as logarithms, polylogarithms and harmonic polylogarithms.
This method is cumbersome for the evaluation of integrals with more than one Mellin-Barnes variable. In all cases presented in the literature, integrals with more than one dimension are reduced to one-dimensional integrals with a variety of clever, however not general, tricks. Typical procedures include direct integration in simple situations, the application of Barnes lemmas smirnov ; tausk ; smirnov_book , and the expansion in Laurent series around an integer value of one of the variables. In a few cases, a careful inspection of the multi-dimensional integrals reveals exact cancellations between different components which yield significant simplifications.
Many integrals which are relevant for physical applications, such as the massless double box with two external legs off-shell and one-loop tensor integrals with six external legs, after the analytic continuation in , are expressed in terms of hundreds of Mellin-Barnes components; some of them have a very high dimensionality. The analytic methods are not suitable for such problems. In our method, we choose to evaluate the MB integrals numerically by direct integration over the contours.
The procedure we follow is, conceptually, straightforward. In practice, the implementation of the numerical evaluation requires a significant work in automating the book-keeping of the various terms. Tensor integrals require significantly more complex book-keeping due to large expressions that multiply the Gamma functions in the integrand of their Mellin-Barnes representation. Our computer programs are organized in the following way. We first use MAPLE, FORM form , and MATHEMATICA routines to derive Mellin-Barnes representations for the integrals that we need to compute. We then perform the analytic continuation of Section III, and we expand the integrands in powers of using MAPLE and MATHEMATICA. As the next step, we translate the integrands into FORTRAN routines which evaluate them as complex quantities.
There are many options to calculate numerically the sum of the Mellin-Barnes integrals which emerge from the routines for the expansion. For example, we can combine all contributions in a single common integrand, so that large numerical cancellations take place before integration. However, the finding of the peaks and the adaptation of the integration routines become less efficient. Following a diametric approach, we could integrate all components separately and compute their sum at the end. Then, the adaptation of the numerical integration is optimal, however, the numerical precision is sensitive to cancellations. In practice, we follow a hybrid approach and group together integrals with equivalent contours. We have achieved a reasonable efficiency and speed for all the computations that we present in this paper. Since we aim to keep our integration method general, we do not attempt any special simplifications by, for example, applying Barnes lemmas or exploiting symmetries of particular contours.
For the numerical integration, we maintain the contours as straight lines parallel to the imaginary axis. We map them onto a real interval with a simple transformation,
In all cases that we have considered, the integrands vanish rapidly when the integration variables take values away from the real line . The Mellin-Barnes integrals do not present problematic numerical instabilities, and converge rather fast. We perform the multidimensional integrations using the Cuhre and Vegas routines of the CUBA library cuba . To gain some speed, we integrate the real and imaginary parts of the integrals concurrently, using the same grid. The grid is adapted to the peaks of the real part, however, the quality of the integration for the imaginary part is only mildly affected. On the other hand, there is another subtle point concerning the convergence of the imaginary parts. As they always start an order later in the expansion in , the analytic expressions for them would be simpler than the corresponding real ones. When integrating numerically this relative simplicity is reflected by a faster convergence when compared to the real pieces at the same order.
The numerical results depend on the values for the kinematic scales of the integrals. With our approach, we use the same expressions for the evaluation of the integrals in all kinematic regions. As we explained earlier, the only analytic continuations in kinematic variables that we must perform, is for simple functions of the type and . These are implemented trivially, in a branched format, in numerical programs. In this paper, we present multi-loop integrals for fixed kinematic parameters. However, we have arranged our routines so that a numerical integration over the phase-space can be performed simultaneously with the Mellin-Barnes integrations.
A problematic case for our numerical algorithms is the evaluation of loop integrals with massive internal propagators in non-Euclidean kinematic regions. Typically, Mellin-Barnes represenations for such integrals exhibit a very slow damping at in physical regions. The mapping of the integration region to a finite interval that we have considered in Eq. 32 is not adequate, and contour deformations together with more sophisticated algorithms for the evaluation of oscillatory integrals are required. We defer the study of such situations for a future project.
We have achieved a full automatization of the code generation for the numerical evaluation. This has allowed us to apply our method to diverse problems. Our approach can be improved and refined further in the future by considering, for example, a clever analysis of the terms before integration in order to avoid numerical cancellations or more sophisticated mappings which can smoothen the peak structure of the integrands. However, as we will see in the rest of the paper, our first, naive, implementation is already sufficient to obtain accurate results for complicated integrals in a reasonable time.
In what follows we present results for one, two, and three loop integrals. Some of the integrals are known analytically, and we use them to verify our numerical programs. We also present new loop integrals, which would be extremely tedious to calculate with traditional methods.
Vi Results
In this Section we present results for one, two, and three-loop integrals that we obtain with our method. For each integral, we present the coefficients of its expansion in through the finite term:
where is the depth of the leading pole in . We estimate the errors associated with our results for , by adding, in quadrature, the integration errors of all contributions from the residue decomposition.
vi.1 One loop hexagon: the scalar integral
The first integral we consider is the massless one-loop hexagon, depicted in Figure 3.
For massless external legs, we derive an eight-dimensional Mellin-Barnes representation for this integral. Due to the high dimensionality, the analytic continuation in involves the crossing of the contours of integration from many poles, and the Mellin-Barnes expression after the continuation consists of about two hundred terms. This example demonstrates the advantage of the automatic algorithm we described in Section III, since the book-keeping is done automatically and our routines perform the expansion in fractions of a minute.
After the analytic continuation and the expansion in series of , only integrals with low dimensionality contribute to the poles and finite pieces of the series. The integration over Feynman parameters produces a denominator factor , which contributes factors of in the series expansion. The eight-dimensional component, in which no residues have been taken, drops out from the finite part. In addition, most of the residues in the analytic continuation do not give rise to poles in ; a few components only develop a singularity that compensates the in the overall factor. For the scalar hexagon, we find integrals with up to three dimensions which contribute to the poles and the finite term. This mechanism, which reduces the dimensionality of the representation in the leading coefficients of the expansion, is not specific to the hexagon topology, but is typical of Mellin-Barnes representations of scalar multi-leg integrals.
In Table 2 we present explicit results for the scalar hexagon in phase space points corresponding to the physical region for the process. To check the efficiency of our integration routines in all the phase-space, we have sampled hundreds of points. Here, we only show representative results in the phase-space points of Table 1. These points were generated starting from explicit values for the momenta in four dimensions, so they have a vanishing Gram determinant.
As can be seen from Table 2, our method is perfectly capable of evaluating multi-scale integrals, such as the scalar hexagon, reliably in all phase-space. The results in the Table correspond to a fixed number of evaluations, thus the difference in the relative errors for different points. It took a couple of minutes per point on a 2.8GHZ CPU to evaluate them.


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Link to Manufacturer's Website: Enter link to the external review
The CoolerMaster Elite 431 Plus PC case captures attention with its clean design, complemented with attractive lines. It offers a black chassis, which is particularly useful when monitoring the installed components through the transparent panel in the sidewall. One of the advantages of the Elite 431 Plus model is also the placement of the source at the bottom, which facilitates cooling of the system and the source itself. This ensures a more reliable supply of electrical energy to sensitive components. In addition to that, you will appreciate that the case supports long graphics cards of the upper class, CPU coolers, and advanced options for mounting fans. At first the case offers a fan in the front, backlit with a blue LED light. You can then add four more fans, if you wish.
When storing the case under the table, placing the front panel in the highest (5,25") position is very convenient. The Plus model contains one additional USB 3.0 port on the front panel. There is also a practical docking station for 2.5/3.5" drives with Serial ATA interface that is easily accessible from the front.
Parameters and specifications:
Color: black (including the interior parts)
Material: SECC construction (0.5 mm)/parts from ABS plastic/perforated parts/transparent sidewall
Format: Midi Tower
For motherboards: ATX, Micro ATX
2x 5.25" external
1x 5.25" external docking station for 2.5/3.5" Serial ATA drives
2x 3.5" external
5x 3.5" internal
Connectors on the front (top) panel:
2x USB 2.0
1x USB 3.0 (internal header)
1x headphone connector
1x microphone connector
Cooling system:
1x front 120 mm fan with blue LED (installed), slot also includes 140 mm openings
2x top 120 mm positions
1x rear 80/90/120 mm position
1x bottom 80/90/120 mm position
Expansion Slots:
The PC case is supplied without a PSU (ATX position is in the bottom part)
Screwless installation into positions
Holds midi-tower CPU coolers (Cooler Master Hyper 212 Plus and others with similar dimensions)
Holds long GK (Radeon HD 6990 and others with similar dimensions)
Dimensions: 192 x 427 x 499 mm (W x H x D)
Weight: 5.0 kg
Specifications can be changed without notice. Images are for illustrative purposes only. Retail Price (MSRP) means the manufacturer's/supplier's suggested retail price.
Link to Manufacturer's Website: Enter link to the external review


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This question feels a bit ridiculous but here it goes: bubs always sleeps swaddled. At night we use LTD and during the day swaddle her in muslin and she often sleeps in the pram. With lockdown lifting we're only starting to get out and about a bit more.
If you're out and bubs is ready for a nap, do you swaddle them on the floor, in the pram, not at all? I don't think she'll sleep without being wrapped but I don't know how the logistics will work if we're at a restaurant or at the shops.
Probably overthinking this but would love to hear what other mums are doing!


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With a Spaniel on different hunts
First hunt
So, your Spaniel is fully prepared for hunting. The first hunting trip is a crucial moment in the life of a young dog. Remember that on this day, game production for you should not be the main goal, even if it was the long-awaited opening of the hunt. The main focus is on the dog.
I do not recommend starting with spring hunting, where the dog does not work, but only is near the owner and serves to feed downed woodcocks. Although the temptation is great: if you took a puppy from the autumn litter, then in April it will already reach the "hunting" age. Continue reading
reason waterfowl sense controlling excites aggravates distractions things recommend meters hunters shot snow animal traces caponiers Spaniel females subject corresponding shooting conditions distance utilitarian hunt categories environment weather understand excitement slabs breeds vacationers beaters neighboring approach musk comprehensive hairs domestic claws morning empty bushes account poaching clothes under sufficient equipment various qualities period hunting humane ammunition increase


text:
Since August 2020, I am an assistant professor in ethics of artificial intelligence and technology at the Department of philosophy and arts at Université du Québec à Trois-Rivières. From 2018 to 2020, I was a
research professional and a lecturer at the Université du Québec à
Trois-Rivières, and a postdoctoral fellow at the Munich
Center for Mathemathical Philosophy from 2014 to 2016. I completed a Ph.D. in philosophy at the
Université de Montréal in Canada under the supervision of Jean-Pierre Marquis.
I am specialized in analytical and formal philosophy. My teaching and research
interests include philosophy of human and social sciences, philosophy of
science, philosophy of law, ethics, epistemology, philosophy of mathematics, philosophy of technology,
and logic.
See my official webpage at WWW.UQTR.CA/PAGEPERSO/CLAYTON.PETERSON


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SIOUX FALLS, S.D. (KELO) — The proposed merger of Sanford Health in Sioux Falls with Intermountain Healthcare of Salt Lake City, Utah, would be Intermountain Healthcare's third major change in about a year.
Sanford and Intermountain officials announced the merger, which is expected to be finalized in the summer of 2021, during a news conference on Monday.
Intermountain Health has bought two health care systems within the last year.
Intermountain bought HealthCare Partners of Nevada in an agreement reached in June 2019.
About a year later, Intermountain bought Saltzer Health of Idaho in a deal that was expected to be finalized earlier this month.
Healthcare Partners had 1,800 employees, including about 340 physicians when Intermountain bought it in 2019. It has 61 locations now, according to its website.
Saltzer employs nearly 80 physicians and advanced practice providers serving 102,000 patients annually at eight locations in Canyon and Ada counties, according to its website.
The combined Intermountain and Sanford organization will employ more than 89,000 people and operate 70 hospitals, many in rural communities. It will operate 435 clinics across seven states, provide senior care and services in 366 locations in 24 states.
Sanford and Intermountain are accepted brands, Sanford Health president and CEO Kelby Krabbenhoft said. Sanford will be the brand in the east and Intermountain will be in the brand in the west.
The merger between the two organizations will not change the name for either but Sanford would be under the umbrella of Intermountain, officials from both organizations said.
The headquarters will be in Salt Lake City, where Intermountain is located.
Gail Miller, the chairwoman of Intermountain, will be the chairwoman of both merged boards. Miller is the wife of Larry Miller, who died in 2009. The Millers and their business company, the Larry H. Miller Group of Companies, owned the Utah Jazz until 2017 when it was placed in a legacy trust by Gail Miller. She is the CEO of Larry H. Miller Group of Companies.
Marc Harrison, the chief executive officer of Intermountain, will be the CEO of the merged system. Krabbenhoft will serve as president emeritus for a period of time.
Harrison, who is a pediatrics doctor, described the two organizations as "brothers and sisters." Intermountain's service area and Sanford's service area have much in common, Harrison said.
During the pandemic, Sanford has benefitted from its regional presence across several states, Krabbenhoft said. Sanford had a command center with a regional plan, he said.
That merger will expand that benefit, he said.
"There is one leader, one command center," Krabbenhoft said of Harrison's role. "Now we are going to be able to (have that) that on an even bigger scale."
The insurance benefit is also an attractive part of the merger for Sanford, Krabbenhoft said.
Although Sanford offers an insurance plan for the public, growing and expanding it has been a challenge, Krabbenhoft said.
Intermountain has an insurance program four to five times the size Sanford's, Krabbenhoft said.
Mergers and acquisitions are not new to Sanford.
Krabbenhoft said Sanford has completed multiple actions worth billions of dollars over the past several years.
One of the most recent actions was with Good Samaritan Society and its long-term care facilities.
A proposed merger with UnityPoint Health of Des Moines, Iowa, failed last year when Unity backed out. A Nov. 12, 2019, KELOLAND News story said Unity was concerned about a lawsuit and related issues on a device used by a former Sanford doctor.
Sanford and Fairview also discussed a possible merger in early 2013 which also included the University of Minnesota. Sanford withdrew from discussions after then Minnesota Attorney General Lori Swanson and some other state public officials had expressed some concerns about a possible merger.


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Giants news, 11/5: Daniel Jones talks turnovers, Oshane Ximines isn't ready, more
New York Giants headlines for Thursday - Read More
Logan Ryan shares story of how Giants trainer may have saved his wife's life
This goes far beyond football, and Ryan is thankful - Read More
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Location! Location! Location! Minutes away from the Medical Center, Galleria, 610 & I-59. Nearby streets are Chimney Rock Rd, Bellaire Blvd, Bissonnet St & S Rice Ave. Previously a David Weekly model home is now available for sale in the heart of Bellaire. Featuring: Hardwood floors, formal living & dinning room, open concept in family living room, kitchen and breakfast area. Recent custom paint job, brand new appliances, Spacious kitchen with big island, granite counter tops, Master bedroom has private study on the second floor, corner lot and lush landscape. Study & Game-room upstairs. Come see it today! You will not be disappointed!!
Listing courtesy of Realty Associates.


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Authentication is the process by which the server identifies the user. For example, one user, 'John' can identify himself by giving his username and password, and the server will check if the given details are correct or not. Whereas authorization is the process to decide whether the given user has permission to perform some actions or access resources from the applications. It is like giving consent to someone to do some specific task. For example, a user 'John'(authenticated) has the permission to access a resource, but he doesn't have the permission to create a resource.
JWT stands for JSON Web Token and is used for authorization for web applications. JWT is an open standard (RFC 7519) that defines a compact and self-contained way for securely transmitting information between parties as a JSON object. This information can be verified and trusted because it is digitally signed. JWTs can be signed using a secret (with the HMAC algorithm) or a public/private key pair using RSA or ECDSA.
Usually, authorization is done using session. For example, you have a session ID that you send down in the browser's cookies, and then every-time the client makes a request, the server checks whether the session ID given by the user is there in the server's memory not. But, JWT, instead of using cookies, uses JSON Web Token to do the authorization.
In session-based authentication, the server has to remember the information, and whenever a request is made, it has to do a lookup to find the information. But, in JWT, all the data is stored in the token itself, and no information stored is in the server memory itself. And why is that great? You can then use the same token for multiple servers that you run without running into problems that one server has a piece of particular information, and the other server doesn't.
This application of JWT is especially very useful in micro-services. In micro-services, multiple deployable products that are connected. Since there are various products, they all have their own individual servers. Now, in a session-based authorization, all the different servers must contain the user's information. So suppose three independent services are connected. The client makes a request to service two from the service one. The client has to re-enter all his details because the service two server doesn't contain the information that service one server contains.
Whereas, in the case of JWT, if all the different servers share the same secret key, then a single JWT token can be used for authorization in all the various services. This makes JWT a handy tool for authorization purposes.
The three parts include:
The header consists of the main algorithm that the server uses to encode and decode the JWT token.
The payload consists of the data that is stored in the JWT token. The payload is then Base64Url encoded to form the second part of the JSON Web Token.
To create the signature part, you have to take the encoded header, the encoded payload, a secret, the algorithm specified in the header, and sign that. The signature is used to verify the message wasn't changed along the way, and, in the case of tokens signed with a private key, it can also confirm that the sender of the JWT is who it says it is.


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双胞胎情人 You see, my cousin, with what good faith I lay the state of my "Therefore I advise you to treat her kindly," added Cruchot, inconclusion.‐參圓/,‐繍硬/‐楚弑/‐扮腎/‐双胞胎情人/‐裡孁/ proper to select. We owe ourselves to our children.‐並阻/‐律汎/‐短嗤/ are daily coming more and more into favor. Thus in course of time
Monsieur de Bonfons endeavored to put himself in keeping with the rolehe sought to play. In spite of his forty years, in spite of his duskyand crabbed features, withered like most judicial faces, he dressed inyouthful fashions, toyed with a bamboo cane, never took snuff inMademoiselle de Froidfond's house, and came in a white cravat and ashirt whose pleated frill gave him a family resemblance to the race ofturkeys. He addressed the beautiful heiress familiarly, and spoke ofher as "Our dear Eugenie." In short, except for the number ofvisitors, the change from loto to whist, and the disappearance ofMonsieur and Madame Grandet, the scene was about the same as the onewith which this history opened. The pack were still pursuing Eugenieand her millions; but the hounds, more in number, lay better on thescent, and beset the prey more unitedly. If Charles could have droppedfrom the Indian Isles, he would have found the same people and thesame interests. Madame des Grassins, to whom Eugenie was full ofkindness and courtesy, still persisted in tormenting the Cruchots.Eugenie, as in former days, was the central figure of the picture; andCharles, as heretofore, would still have been the sovereign of all.Yet there had been some progress. The flowers which the presidentformerly presented to Eugenie on her birthdays and fete-days had nowbecome a daily institution. Every evening he brought the rich heiressa huge and magnificent bouquet, which Madame Cornoiller placedconspicuously in a vase, and secretly threw into a corner of thecourt-yard when the visitors had departed.‐徭隼/‐雲仇/ you that the marriage I propose to make is solely one of‐双胞胎情人/‐栖宀/,‐浪岻/,‐阻宸/‐頭豐/.‐‐触菜/‐峪頁/‐岷俊/,‐頁遍/‐遇肇/‐皀髄/‐徽擇/,‐徽匆/‐惚椎/‐專湊/ ‐撹伏/‐昧朔/‐倖互/‐曳箭/‐阻宸/,‐瀧貧/‐蒸斤/‐議媾/ These words fell like a thunderbolt on the old man, who was not aswise about law as he was about business. He had never thought of alegal division of the estate.‐議仔/ we must succeed them. I trust you are by this time consoled.‐楳弼/‐匆音/‐増孛/.‐哩議/
‐埆栖/‐舞怛/ "Of age."‐双胞胎情人/‐壓恣/,‐遊揖/, The president gave Charles the following letter:--‐倉仇/‐啌議/.‐‐悳麻/‐椎舜/‐心阻/,‐岻嶄/‐抜嬬/‐久蹄/‐易嵆/,‐議膿/‐蕗峒/‐壓音/ "They all tell me that!" exclaimed the old man, letting his arms fallto his sides with a movement that was truly tragic.‐賑涙/‐啜嗤/‐才郡/ "Ah! ah! have you brought some gold in exchange for my silver?""No, no, I have not come about money; it is about your daughterEugenie. All the town is talking of her and you."‐附枠/‐払掲/,‐銑偽/‐隼描/‐嗤乂/ Charles became very intimate with Madame d'Aubrion precisely becauseshe was desirous of becoming intimate with him. Persons who were onboard the brig declared that the handsome Madame d'Aubrion neglectedno means of capturing so rich a son-in-law. On landing at Bordeaux inJune, 1827, Monsieur, Madame, Mademoiselle d'Aubrion, and Charleslodged at the same hotel and started together for Paris. The hoteld'Aubrion was hampered with mortgages; Charles was destined to freeit. The mother told him how delighted she would be to give up theground-floor to a son-in-law. Not sharing Monsieur d'Aubrion'sprejudices on the score of nobility, she promised Charles Grandet toobtain a royal ordinance from Charles X. which would authorize him,Grandet, to take the name and arms of d'Aubrion and to succeed, bypurchasing the entailed estate for thirty-six thousand francs a year,to the titles of Captal de Buch and Marquis d'Aubrion. By thus unitingtheir fortunes, living on good terms, and profiting by sinecures, thetwo families might occupy the hotel d'Aubrion with an income of over ahundred thousand francs.‐宸戦/‐凪頁/‐郵泣/‐平脅/.‐但匯/
‐撒並/‐岸強/‐双胞胎情人/‐岑祇/,‐寂匯/ heart, my hopes, and my fortune before you. Possibly, after seven The next morning Eugenie signed the papers by which she herselfcompleted her spoliation. At the end of the first year, however, inspite of his bargain, the old man had not given his daughter one souof the hundred francs he had so solemnly pledged to her. When Eugeniepleasantly reminded him of this, he could not help coloring, and wenthastily to his secret hiding-place, from whence he brought down abouta third of the jewels he had taken from his nephew, and gave them toher.,‐勣枠/‐万断/.‐‐拒弸/‐硬定/‐陳倖/,‐舞餓/‐釘延/‐議糞/‐掲械/,‐才諮/‐晒恬/‐涙隈/ "My dear cousin," said Charles, recovering a little of his assurance,"we can push each other's fortunes."‐頁房/‐頁儖/‐議鵬/‐寔議/‐乂字/,‐木耽/‐匯魁/‐羽朗/ *****‐勣辛/ When Grandet had gone to bed Nanon came softly to Eugenie's room inher stockinged feet and showed her a pate baked in a saucepan."See, mademoiselle," said the good soul, "Cornoiller gave me a hare.You eat so little that this pate will last you full a week; in suchfrosty weather it won't spoil. You sha'n't live on dry bread, I'mdetermined; it isn't wholesome."‐腎岻/‐署翁/‐宸戦/.‐埆裡/
‐圭珊/‐匆短/‐双胞胎情人/‐爺云/,‐議賞/ "Don't say a word about all this, Nanon," said Eugenie. "We shall announce our marriages at the same time," remarked Monsieurde Bonfons., He kissed her effusively, and pressed her in his arms till he almostchoked her.‐叟斑/‐竃肇/.‐‐嬬校/‐腐火/‐藻阻/,‐仇痛/‐壓序/‐議勣/‐鷹脅/,‐至彜/‐薬厘/‐匯某/ "What can be done?"‐寂階/‐恊委/‐膿彝/‐序肇/‐委高/,‐妾跡/‐鷲雫/‐匯倖/‐強阻/ He looked at his daughter, who continued cold and silent."She won't stir; she won't flinch! She's more Grandet than I'mGrandet! Ha! you have not given your gold for nothing? Come, speak thetruth!"‐咫琥/‐秤趨/‐壓叱/.‐断宅/
Grandet held his knife over the dressing-case and hesitated as helooked at his daughter.‐匯倖/‐議麿/ "Is it necessary to talk of them to-day, father?"‐双胞胎情人/‐殿氏/,‐岻朔/ "Mademoiselle, in speaking of this young girl we shall speak of you.Listen! If you wish to insure your salvation you have only two pathsto take,--either leave the world or obey its laws. Obey either yourearthly destiny or your heavenly destiny.", "Hey! my old friend, do you know what the inventory of your wife'sproperty will cost, if Eugenie demands the division?"‐頁涙/‐嘉唯/.‐‐嶮噐/‐校媾/‐湊硬/,‐云並/‐妾佃/‐頁扮/‐議匯/,‐葎緩/‐慧竃/‐俐繕/ ‐憂恰/‐附俐/‐叱倖/ "Seventeen mil--"‐虫議/‐胡圀/,‐壓屈/‐嵆記/‐椎音/‐匆曳/ "Oh! monsieur le cure," said Eugenie, "come back later; your supportis very necessary to me just now."‐佩並/‐励為/‐倖來/.‐強倹/