Quaternary Quadratic Forms: Computer Generated Tables

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The P-t curves and quantitative thermo-kinetic parameters had obvious changes in high concentration, such as the peak of the second exponential growth phase reducing, the time of peak postponing, the slope of curves diminishing and the area of curves lowering. Besides, curves and parameters changed with the concentration increasing. All these demonstrated that the growth of E. The P-t curves of E. The thermo-kinetic parameters from the P-t curves of the growth of E.

The influence of flavonoids on the growth of E. The P-t curve of E. They both obey the following equation. In this equation, P 0 and P t are the power at time 0 and t. Using this equation, the quantitative thermo-kinetic parameters such as the power of the first and second peak P 1 and P 2 represent the strength of E. Coli metabolism. The growth rate constants k 1 and k 2 of the first and second exponential phase for E. The heat outputs in stage 1 and stage 2 Q 1 and Q 2 are obtained from the P-t curve of E. Eight thermo-kinetic parameters P 1 , P 2 , t 1 , t 2 , Q 1 , Q 2 , k 1 , and k 2 which profile the metabolism of E.

Obviously, these parameters changes were in line with the concentration of flavonoids. It is necessary to find out the main parameter s playing crucial role in evaluating the anti-bacterial effects, but the key information reflected from these changes is hard to figure out. Hopefully, PCA could solve these problems well. A component correlation matrix of the principal components Table 9 was obtained by PCA analysis. The component correlation of each factor is equivalent to the coefficient of the factor normalized in the principal component equation.

The equation of main component was formed according to this matrix. Each quantitative parameter is normalized in this equation. The equation indicated that P 2 , k 1 , and k 2 might be the main parameters playing more important role in evaluating the anti-bacterial effects of D. By further comparison of k 1 and k 2 from the equation, we found that k 2 contributed more than k 1 to PCA.

Where k 0 refers to the growth rate constant of the control, k c refers to the growth rate constant in the second exponential growth phase at an inhibitor concentration of c. The Probit regression with SPSS statistical analysis software was conducted to calculate the half-inhibitory concentration IC 50 after antibacterial rate counted. IC 50 representing the sensitivity of bacteria to flavonoids was one of the most important indexes in evaluating the anti-bacterial activity of flavonoids.

External standard calibration lines were generated by injection of standard solutions at five concentrations in triplicate, plotting the peak area y obtained from the UPLC analysis against the concentration x of the standard material displaying a linear regression with correlation coefficient r calculated out. The peak of flavonoids was characterized by large areas and good segregation from consecutive peaks.

Chromatograms of S1—S3 under different wavelength. Interestingly, the Cianidanol was most abundant among the four flavonoids. The spectrum of S1—S3 under nm demonstrated that the retention time of Quercetin Q was While that of Quercetin in S3 was The total flavonoids for S1-S3 were Based on the IC 50 of sample, S2 exhibited favorable anti-bacterial effect. The analysis of variance table for quadratic general rotary unitized design. The regression equations of antibacterial rate were obtained from statistical model of quaternary quadratic general rotary unitized design.

Quadratic forms and their applications. Proc. Conf

That was. The mathematic model with good fitting degree could reflect the experimental file from the two aspects. Hence, the parameters calculated from the model had favorable confidence level. The four flavonoids had independent anti-bacterial effect with Luteolin and Cianidanol showing the main and better anti-bacterial effect from the analysis of variance table. The diagram of single factor effect for S1—S3 was rationally calculated from the rotational combination design model.

The single factor under different code level was predicted to make out the antibacterial rate Y-value with others factors as 0-code in this method. The effect of different factors on bacteria could be observed via the range of each factor the difference of Y-value for the highest code between the lowest one and change tend single factor in the analysis of single factor effect. The result showed that X 1 Luteolin mainly contributed to the antibacterial rate Y-value with X 3 Cianidanol and X 4 Quercetin contributing only marginally.

Unexpectedly, X 2 L-Epicatechin had little influence on antibacterial rate. Analysis of the effect of two-factor interaction that defined other factor as 0-code for the four flavonoids was obtained from the statistical model of quaternary quadratic general rotary unitized design. The changes of different factors on Y-value as well as the relationship between two factors could be found out via Y-value of two factors under different levels of code by mathematical model. The analysis of two factors interaction effect diagram.

The diagram of two-factor interaction effect for X 1 Luteolin and X 3 Cianidanol was rationally calculated from the rotational combination design model. When X 1 and X 3 factor were in high code level, the synergistic activity was obvious between X 1 and X 3 with the antibacterial rate Y-value in the top level, which suggested that there was a positive effect between high concentration of X 1 Luteolin and X 3 Cianidanol and antibacterial rate Y-value.

With the decline of code of Luteolin and Cianidanol, Y-value antibacterial rate decreased rapidly with a smaller degree. Therefore, we predicted that there existed positive and synergistic effect between Luteolin and Cianidanol. When Luteolin and Cianidanol was in a higher concentration, there existed obvious synergistic effect between them.

But when Luteolin and Cianidanol in low concentration, no marked synergistic effect was observed, even presenting slight antagonistic effects similar with the relationship of X 1 Luteolin and X 4 Quercetin. Obviously, the analysis showed that there existed both single factor effect and interaction among two factors. So it was difficult to find out the best antibacterial concentration and combinations from the results of the single factor effect and interaction analysis, and quaternary quadratic regression mathematical model did not have the maximum value of bacterial inhibition rate.

Code value of factors was that X 1 : 0. In other words, Luteolin was The actual antibacterial rate was Finally, with the use of validated mathematical model, we finally obtained the combination that Luteolin was 1, To verify the result that antibacterial activity of the four major components from the leaves of D. The result showed that Luteolin In the final calculation, only Probit regression can be used to predict the IC 50 of L-Epicatechin The exact IC 50 of L-Epicatechin remains to be further analyzed yet, but that the antibacterial activity of L-Epicatechin is much weaker than that of Luteolin, and Cianidanol and Quercetin can be confirmed.

By comparing the IC 50 values, we can confirm that antibacterial activity of the four major components from the leaves of D. In the quadratic general rotary unitized design model, we analyzed two factors interaction effect: Luteolin and Cianidanol, Quercetin, and Luteolin had synergistic effects on antibacterial activity at high doses.

In order to verify this result, we independently measured the antimicrobial activity of the two combinations. The result showed that the IC 50 values of the combination of Luteolin and Cianidanol was The ratio of their anti-microbial activity P in IC 50 value through the concentration ratio weighted average method based on the IC 50 values of the four flavonoids alone and two combinations of two flavonoids was calculated.

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The results displayed that the IC 50 value for the combination of Luteolin and Cianidanol was increased to 1. Escherichia coli , a Gram negative bacteria, is common and important to screen flavonoids with anti-bacterial activity as a model organism Sondi and Salopek-Sondi, In the study, we investigated the anti- E.

According to the P-t curves of E. The previous study has demonstrated that flavonoids are abundant in the leaves of D. Ethanol extract from Mentha longifolia containing the components of Luteolin, Apigenin, Quercetin, and Kaempferol was the most active fraction against the tested bacteria Akroum et al. The combination of Luteolin and Amoxicillin displayed synergistic activity against E. Furthermore, the novel Luteolin derivatives showed favorable antibacterial activity in vitro against B.

To be specific, the content of Luteolin in S2 was The content of Quercetin in S2 was However, the content of flavonoids in extracts did not show a linear relationship with the IC 50 value, indicating that there existed unknown compounds with significant antibacterial activity from the leaves of D. Orlov, D. Parimala, R. Hautes Etud.

Pfister, A.

Expressing a quadratic form with a matrix

Notes , Cambridge University Press Prestel, A. Prestel, A; Ware, R. Rost, M.

Quaternary Quadratic Forms - Computer Generated Tables | Gordon L. Nipp | Springer

Scharlau, W. Szyjewski, M. Algebra Anal. English translation: Leningrad Math. Tignol, J. A 42 — Vishik, A. Voevodsky, V. Wadsworth, A. Witt, E. The first nontrivial result in this direction is due to J. All known results make natural the following conjecture. The first-named author was supported by Alexander von Humboldt Stiftung.

The first-named author died on 17th April, In this paper, we give only a partial answer to the conjecture. Forms with maximal splitting. It turns out that Conjecture 1. Let us recall some basic definitions and results. In [4] Hoffmann proved the following Theorem 1. Conjecture 1. The following example is due to Hoffmann. Let F be the field of rational functions k x1 ,. Then q has maximal splitting and is not a Pfister neighbor.

This example gives rise to the following Proposition 1. Let us return to Problem 1. By Proposition 1. Plan of works. In section 3, we prove Theorem 1. Our proof is based on the following ideas of Bruno Kahn [11] : First of all, we recall some result of M. Knebusch: let q be an anisotropic F -form. If the F q -form qF q an is defined over F , then q is a Pfister neighbor. Now, let q be an F -form satisfying the hypotheses of Theorem 1. This completes the proof of Theorem 1. To prove Theorem 1.

Definition 1. The final step in the proof of theorem 1. Theorem 1. Let Q be the projective quadric corresponding to an anisotropic F -form q. Assume that Q admits a Rost projector. The following statement is an evident corollary of the theorems above. Corollary 2. Remark 2. Actually, the restriction on the integer n here is unnecessary at least in characteristic 0 - see Theorem 7. For this reason, we use the notation Z for all groups and rings Z throughout the text. In section 5 we work in the classical Chow-motivic category of Grothendieck see [3],[20],[31],[29].

Laghribi [18]. Namely, we reduce Theorem 1. As in the paper of B. Kahn, we work modulo a suitable power I n F of the fundamental ideal I F. We start with the following notation. Definition 3. By Lemma 3. The following lemma is an obvious generalization of this statement. Lemma 3. Let A be a central simple F -algebra of index 2n and q be a quadratic form over F. We start with the following case: Case 1. By Corollary 2. This completes the proof in Case 1. Case 2. After this, Lemma 3.

By Corollary 3. We get a contradiction. The proof is complete. Theorem 3. Then the following conditions are equivalent. By Proposition 3. By Theorem 3. Elementary properties of AMS-forms In this section we start studying forms with absolutely maximal splitting AMSforms defined in the introduction Definition 1. Lemma 4. Let X, Y and Z be smooth projective varieties over k of dimensions l, m and n, respectively.

Clearly, it is enough to consider the second possibility. P Lemma 5. Binary direct summands in the motives of quadrics The following result was proven but not formulated by the second author in his thesis see the proof of Statement 6. Proof of Theorem 6. The construction we use here is very close to that used by V. Voevodsky in [36]. We will denote its motive by XP. By Theorem A. From this point, we will denote M pr simply as pr since we will not use simplicial schemes themselves anymore. Sublemma 6. This, combined with Sublemma 6. From Sublemma 6. But, by Sublemma 6.

Hence the second is zero as well. But by Theorem A. It follows from Theorem 6. By Sublemma 6. Then, by Theorem A. By Lemma 6. We begin with the following modification of Theorem 1. Theorem 7. If an anisotropic quadratic form has absolutely maximal splitting, then it is a Pfister neighbor. In the proof of Theorem 1. We need also the following easy consequence of a result by Hoffmann. Lemma 7. Obvious in view of [4, Lemma 5]. Then Theorem 7.

Voevodsky, which we use in the proof of Theorems 6. First of all, we need some facts about triviality of motivic cohomology of smooth simplicial schemes. In the case of a smooth variety we have further restrictions on motivic cohomology: Theorem A. In [36] the Stable homotopy category of schemes over Spec k , SH k was defined see also [25]. In [36], Section 3. Theorem A. HM i Let P be some smooth projective variety over Spec k. Remark A. Actually, Theorem 4. At the same time, Theorem A. One should observe that the same proof as in [36, Lemma 3.

The following result of V. Voevodsky is the main tool in studying motivic cohomology of quadrics: Theorem A. In [36, Lemma 4. We should add that in [36, Theorem 4.


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References [1] Arason, J. Cohomologische invarianten quadratischer Formen. Algebra 36 , no. Unramified Witt groups of real anisotropic quadrics. Ktheory and algebraic geometry: connections with quadratic forms and division algebras Santa Barbara, CA, , —, Proc. Intersection Theory.

Springer-Verlag, Isotropy of quadratic forms over the function field of a quadric. Splitting patterns and invariants of quadratic forms. Splitting patterns of quadratic forms. Quadratic forms with maximal splitting. Algebra i Analiz, vol. Petersburg Math J. Fields of u-invariant 9. Preprint, Bielefeld University, , Degree four cohomological invariants for quadratic forms, Invent.

A descent problem for quadratic forms. Motivic cohomology of smooth geometrically cellular varieties. Unramified cohomology of quadrics I. Motivic cohomology and unramified cohomology of quadrics, J. Characterization of minimal Pfister neighbors via Rost projectors. Generic splitting of quadratic forms, I.

Soc 33 , 65— Generic splitting of quadratic forms, II. Soc 34 , 1— The Algebraic Theory of Quadratic Forms. Massachusetts: Benjamin revised printing: Correspondences, motifs and monoidal transformations. The norm residue homomorphism of degree 2 in Russian , Dokl. Nauk SSSR , English translation: Soviet Mathematics Doklady 24 , On the norm residue homomorphism for fields, Mathematics in St.

Petersburg, , A. The norm residue homomorphism of degree 3 in Russian , Izv. Nauk SSSR 54 , USSR Izv. Algebraic K-theory and quadratic forms, Invent. A1 -homotopy theory of schemes. Preprint in preparation. Hilbert theorem 90 for K3M for degree-two extensions, Preprint, Regensburg, Some new results on the Chow-groups of quadrics. The motive of a Pfister form. Classical motives. The fifth invariant of quadratic forms in Russian , Algebra i Analiz 2 , Integral motives of quadrics. Triangulated category of motives over a field, K-Theory Preprint Archives, Preprint 74, see www.

Milnor conjecture. Izmailov Abstract. Some particular answers to the following question are presented: does a surjective quadratic mapping between Banach spaces have a bounded right inverse or not? These notes are concerned with some particular answers to one question connected with quadratic mappings.

Let X and Y be linear spaces. To say it in other words, a quadratic mapping is a homogeneous of degree 2 polynomial mapping. In the case of a finite-dimensional Y , a quadratic mapping is a mapping whose components are quadratic forms. Recall that for any quadratic mapping Q, there exists a unique symmetric bilinear mapping B related to Q in the sense mentioned above. Hence in the sequel, we shall denote quadratic mapping and associated symmetric bilinear mapping by the same symbol.

Several mathematicians, e.


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Aron, B. Cole, S. Dineen, T. Gamelin, R. Gonzalo, J. Jaramillo, R. Ryan, have studied polynomials in Banach spaces. See the book of Dineen [D], especially chapters 1 and 2, for more information including a comprehensive list of references. When studying singular points of smooth nonlinear mappings, quadratic mappings analysis is of great importance. Let us mention two different and rather productive approaches to the study of irregular problems which have been actively developed for the last two decades.

The first approach is based on the so-called 2-normality concept [Ar1], the other one is based on the 2-regularity construction see [IT] and the bibliography there. But for any approach of such a kind it is typical that second derivatives are taken into account. When the first derivative is onto that is the regular case , the first derivative is a good local approximation to the mapping under consideration.

Applying the highly developed theory of linear operators to linear approximation one can obtain the most important facts of nonlinear analysis, such as the implicit function theorem and its numerous corollaries. Primary 47H60; Secondary 58C The author also thanks IMPA, where he was a visiting professor during the completion of this paper. IZMAILOV Naturally, when the first derivative is not onto that is the singular case , linear approximation is not enough for a description of the nonlinear mapping local structure, and one has to take into account the quadratic term of the Taylor formula.

On the other hand, quadratic mappings are of great interest by themselves because a quadratic mapping is the most simple model of a substantially nonlinear mapping. When we are to construct an example for some theorem on singular points, we consider quadratic mappings first of all. But in contrast with the theory of linear operators, quadratic mappings theory is not really developed so far.

There are no answers to some basic questions, and here we would like to discuss one such question connected with the existence of a bounded right inverse to a quadratic mapping. For linear operators, a complete answer to this question is given by the classical Banach open mapping theorem. Let us recall this result. Now let us replace here the continuous linear operator A by a continuous quadratic mapping Q and consider the same question all the notions and notations remain without any modification. In topological terms the question is: determine the conditions under which the image of a neighborhood of zero element in X is a neighborhood of zero element in Y?

First, let us mention that in contrast with theory of linear operators, in the general case the right inverse to Q can be unbounded when Q is onto, as illustrated by the following example.

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We next consider some sufficient conditions for boundedness of a right inverse, but for that purpose we need some more terminology. Recall that Q here is the symmetric bilinear mapping associated with the quadratic mapping under consideration. Note that N Q is always a closed cone but normally this cone is not convex. Note also that all this terminology can be applied to a general nonlinear mapping at a fixed point, and this results in the basic notions of 2-regularity theory for that purpose one has to consider the invariant second differential of the mapping at the point under consideration as Q.

Some necessary and sufficient conditions for strong 2-regularity of quadratic mappings were proposed in [Ar2]. Let X and Y be Banach spaces. Certainly, conditions 1 and 2 cannot be satisfied simultaneously. If the condition 2 holds, the assertion of Theorem 1 is fairly standard as it follows immediately from the standard implicit function theorem note that in this case Q is surjective automatically, so one does not have to assume that Q is surjective.

The assertion corresponding to condition 1 is not so standard as it follows from a special theorem on distance estimates for strongly 2-regular mappings. Now let us consider the finite-dimensional case. Clearly, in this case the strong 2-regularity condition in Theorem 1 can be omitted.

Note that Example 1 has a strong infinite-dimensional specificity. We do not know any finite-dimensional example of a surjective quadratic mapping with unbounded right inverse, and we have to admit that we do not know so far if such an example is ALEXEY F. However, we would like to discuss some particular answers to this question. We are in an algebraic situation now, and it seems reasonable that in order to find to find such answers, one has to take into account both analytical and algebraic arguments.

To begin with, note that for the finite-dimensional case Theorem 1 without the strong 2-regularity assumption provides a complete description for a typical quadratic mapping. Proposition 1. For any positive integers n and m, the set of 2-regular quadratic mappings is open and dense in the set of all quadratic mappings from Rn to Rm. The set of all quadratic mappings from Rn to Rm is considered here with the standard norm topology. It is well known that this set has a natural linear structure and can be normed in a natural way.

It was stated in [Ag] in a different form. We see now that for almost any finite-dimensional quadratic mapping, the following alternative is true: the null-set is trivial, or the mapping is 2-regular with respect to any nonzero element from the null-set. Hence by Theorem 1, for almost any finite-dimensional quadratic mapping the following implication holds: if it is surjective then its right inverse is bounded.

The answer turns out to be positive for two forms as well, but this case is already not trivial at all. Proposition 2. The first case is important in view of constrained optimization problems. Some special results for this case were obtained by Agrachyov in [Ag]. Let us discuss here the second case. Proposition 3. Let us take advantage of the following procedure which is sometimes useful for a reduction to lower dimension. Q cannot be surjective the subspace Y2 contains elements which do not belong to Q R2.

Now we have only to prove that the equality 4 is contradictory. It is interesting that Propositions 2 and 3 hold true only for quadratic mappings. For polynomial mappings which are homogeneous of degree greater than two, similar results are not obtained. Moreover, we provide an example of a mapping of degree 5 which is onto, but its right inverse is not bounded. Example 2. One can show that the image of a neighborhood of zero element in the original space is not a neighborhood of zero element in the image space here; at the same time, Q is onto.

The easiest but of course, not accurate way to see this is to draw the image of the unit square e. Clearly, these notes contain more questions than answers. A closely related question is: under what assumptions is it the case that not only is a given quadratic mapping surjective but any quadratic mapping close to it is surjective?

In [Ag], a quadratic mapping with such a property was referred to as substantially surjective. There is a good reason to think that a complete answer to one of the questions in these notes will result in a complete answer to another, and perhaps, to numerous important questions about the topology and algebra of quadratic mappings.

The author would like to thank the organizers of QF99 for their hospitality and for financial aid they provided for his participation. The author is also grateful to two referees for their helpful comments. References [Av1] Y. Phys 25 , 24— Avakov, Theorems on estimates in the neighborhood of a singular point of a mapping, Math. Notes 47 , — Geometriya 26 , 85— Arutyunov, Extremum conditions.

Abnormal and degenerate cases in Russian , Factorial, Moscow, Arutyunov, On the theory of quadratic mappings in Banach spaces, Soviet Math. Belash and A. Izmailov and A. Tretyakov, Factoranalysis of nonlinear mappings in Russian , Nauka, Moscow, Physics 24 , — Kearton Abstract. The purpose of this survey article is to show how quadratic and hermitian forms can give us geometric results in knot theory.

In particular, we shall look at the knot cobordism groups, at questions of factorisation and cancellation of high-dimensional knots, and at branched cyclic covers. Contents 1. The Seifert Matrix 2. Blanchfield Duality 3. Factorisation of Knots 4. Cancellation of Knots 5. Knot Cobordism 6. Branched Cyclic Covers 7. Spinning and Branched Cyclic Covers 8. Concluding Remarks References 1. In the smooth case the embedded sphere S n may Mathematics Subject Classification. Key words and phrases. Two such pairs are to be regarded as equivalent if there is an orientation preserving smooth or PL homeomorphism between them.

Then K is the exterior of the knot k, and since a regular neighbourhood of S n is unique up to ambient isotopy it follows that K is essentially unique. Every classical knot k is the boundary of some compact orientable surface embedded in S 3. Consider a diagram of k. Starting at any point of k, move along the knot in the positive direction. At each crossing point, jump to the other piece of the knot and follow that in the positive direction.

Figure 1. Now start somewhere else, and continue until the knot is exhausted. The Seifert circuits are disjoint circles, which can be capped off by disjoint discs, and joined by half-twists at the crossing points. Hence we get a surface V with To see that V is orientable, attach a normal to each disc using a right hand screw along the knot. Note that in passing from one disc to another the normal is preserved. Thus there are no closed paths on V which reverse the sense of the normal: hence V is orientable.

L Proof. Trefoil knot As an example we have a genus one Seifert surface of the trefoil knot in Figure 1. Let u, v be two oriented disjoint copies of S 1 in S 3 and assign a linking number as follows. Span v by a Seifert surface V and move u slightly so that it intersects V transversely. Two simple examples are indicated in Figure 1.

The linking number The two copies of S 1 do not have to be embedded: the general definition is in terms of cycles and bounding chains. L2g Definition 1. Let x1 ,. We see from Table 1. In the second case, the procedure is reversed. Ambient surgery i U Figure 1. For a given knot k, any two Seifert surfaces are related by a sequence of ambient surgeries. Let A be a Seifert matrix. An elementary S-equivalence on A is one of the following, or its inverse. Two matrices are S-equivalent if they are related by a finite sequence of such moves. Any two Seifert matrices of a given knot k are S-equivalent.

After Proposition 1. Consider the diagram in Figure 1. L2g Proof. Recall that if x1 ,. In [36, Theorem 3] the classification of simple knots is completed. The hermitian pairing is due to R. Blanchfield [9]. The formula given here was discovered independently in [22, 44].

The following two results are proved in [22, 23, 44, 45]. In [43, pp ] Trotter proves the following. If A is a Seifert matrix, then A is S-equivalent to a matrix which is non-degenerate; that is, to a matrix with non-zero determinant.

Computer Generated Tables

Of course, det A is the leading coefficient of the Alexander polynomial of k. Definition 2. The following result is proved in [44, p] Theorem 2. The pair MA , , determines and is determined by the S-equivalence class of A. Factorisation of Knots If we have two classical knots, there is a natural way to take their sum: just tie one after another in the same piece of string. Alternatively, we can think of each knot as a knotted ball-pair and identify the boundaries so that the orientations match up.

The latter procedure generalises to higher dimensions. Choose i. Schubert showed in [40] that every knot factorises uniquely as a sum of irreducible knots. The following example is contained in [1], although presented here in a slightly different way. This corresponds to a hermitian pairing as in Theorem 2. By Theorem 2. There is another method, due to J. There are further results on this topic in [18, 17, 19].

I should mention that the method of [25] relies on the signature of a smooth 3-knot being divisible by 16, and hence does not generalise to higher dimensions. The work of Hillman in [20] shows that the classification theorems 1. Example 4. We take R4n to denote the euclidean space, and let e1 ,. Moreover 4. II, Proposition 6. Limitations of space preclude publication of more than this in printed form. A printed appendix through discriminant and for discriminants and follows. The complete tables and appendix through discriminant are compressed onto the accompanying 3. Documentation is included in the Introduction.

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