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Journal articles on the topic 'Mathematics, formulae'

Author: Grafiati

Published: 4 June 2021

Last updated: 30 July 2024

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1

Emel’yanov, Dmitry. "Algebras of Binary Isolating Formulas for Tensor Product Theories." Bulletin of Irkutsk State University. Series Mathematics 41 (2022): 131–39. http://dx.doi.org/10.26516/1997-7670.2022.41.131.

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Algebras of distributions of binary isolating and semi-isolating formulae are derived objects for a given theory and reflect binary formula relations between 1-type realizations. These algebras are related to the following natural classification questions: 1) for a given class of theories, determine which algebras correspond to theories from that class, and classify those algebras; 2) classify theories from the class according to the isolating and semi-isolating formulae algebras defined by those theories. The description of a finite algebra of binary isolating formulas unambiguously implies the description of an algebra of binary semi-isolating formulas, which makes it possible to trace the behavior of all binary formula relations of a given theory. The paper describes algebras of binary formulas for tensor products. The Cayley tables are given for the obtained algebras. Based on these tables, theorems are formulated describing all algebras of binary formulae distributions for tensor multiplication theory of regular polygons on an edge. It is shown that they are completely described by two algebras

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2

Chen, J. T., J. W. Lee, S. K. Kao, and Y. T. Chou. "Construction of a curve by using the state equation of Frenet formula." Journal of Mechanics 37 (2021): 454–65. http://dx.doi.org/10.1093/jom/ufab014.

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Abstract In this paper, the available formulae for the curvature of plane curve are reviewed not only for the time-like but also for the space-like parameter curve. Two ways to describe the curve are proposed. One is the straight way to obtain the Frenet formula according to the given curve of parameter form. The other is that we can construct the curve by solving the state equation of Frenet formula subject to the initial position, the initial tangent, normal and binormal vectors, and the given radius of curvature and torsion constant. The remainder theorem of the matrix and the Cayley–Hamilton theorem are both employed to solve the Frenet equation. We review the available formulae of the radius of curvature and examine their equivalence. Through the Frenet formula, the relation among different expressions for the radius of curvature formulae can be linked. Therefore, we can integrate the formulae in the engineering mathematics, calculus, mechanics of materials and dynamics. Besides, biproduct of two new and simpler formulae and the available four formulae in the textbook of the radius of curvature yield the same radius of curvature for the plane curve. Linkage of centrifugal force and radius of curvature is also addressed. A demonstrative example of the cycloid is given. Finally, we use the two new formulae to obtain the radius of curvature for four curves, namely a circle. The equivalence is also proved. Animation for 2D and 3D curves is also provided by using the Mathematica software to demonstrate the validity of the present approach.

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3

Fazly, Mostafa, and Juncheng Wei. "On stable solutions of the fractional Hénon–Lane–Emden equation." Communications in Contemporary Mathematics 18, no. 05 (July 18, 2016): 1650005. http://dx.doi.org/10.1142/s021919971650005x.

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We derive monotonicity formulae for solutions of the fractional Hénon–Lane–Emden equation [Formula: see text] when [Formula: see text], [Formula: see text] and [Formula: see text]. Then, we apply these formulae to classify stable solutions of the above equation.

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4

Emel’yanov, D. Yu. "Algebras of Binary Isolating Formulas for Theories of Root Products of Graphs." Bulletin of Irkutsk State University. Series Mathematics 37 (2021): 93–103. http://dx.doi.org/10.26516/1997-7670.2021.37.93.

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Algebras of distributions of binary isolating and semi-isolating formulas are derived objects for given theory and reflect binary formula relations between realizations of 1-types. These algebras are associated with the following natural classification questions: 1) for a given class of theories, determine which algebras correspond to the theories from this class and classify these algebras; 2) to classify theories from a given class depending on the algebras defined by these theories of isolating and semi-isolating formulas. Here the description of a finite algebra of binary isolating formulas unambiguously entails a description of the algebra of binary semi-isolating formulas, which makes it possible to track the behavior of all binary formula relations of a given theory. The paper describes algebras of binary formulae for root products. The Cayley tables are given for the obtained algebras. Based on these tables, theorems describing all algebras of binary formulae distributions for the root multiplication theory of regular polygons on an edge are formulated. It is shown that they are completely described by two algebras.

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5

Netz, Reviel. "Linguistic formulae as cognitive tools." Pragmatics and Cognition 7, no. 1 (1999): 147–76. http://dx.doi.org/10.1075/pc.7.1.07net.

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Ancient Greek mathematics developed the original feature of being deductive mathematics. This article attempts to give a (partial) explanation f or this achievement. The focus is on the use of a fixed system of linguistic formulae (expressions used repetitively) in Greek mathematical texts. It is shown that (a) the structure of this system was especially adapted for the easy computation of operations of substitution on such formulae, that is, of replacing one element in a fixed formula by another, and it is further argued that (b) such operations of substitution were the main logical tool required by Greek mathematical deduction. The conclusion explains why, assuming the validity of the description above, this historical level (as against the universal cognitive level) is the best explanatory level for the phenomenon of Greek mathematical deduction.

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6

Bouzeffour, Fethi, and Mubariz Garayev. "Multiple big q-Jacobi polynomials." Bulletin of Mathematical Sciences 10, no. 02 (May 19, 2020): 2050013. http://dx.doi.org/10.1142/s1664360720500137.

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Here, we investigate type II multiple big [Formula: see text]-Jacobi orthogonal polynomials. We provide their explicit formulae in terms of basic hypergeometric series, raising and lowering operators, Rodrigues formulae, third-order [Formula: see text]-difference equation, and we obtain recurrence relations.

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7

Markhabatov, N. D., and S. V. Sudoplatov. "Pseudofinite Formulae." Lobachevskii Journal of Mathematics 43, no. 12 (December 2022): 3583–90. http://dx.doi.org/10.1134/s1995080222150215.

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8

Franjic, I., and J. Pecaric. "On corrected Bullen-Simpson's $3/8$ inequality." Tamkang Journal of Mathematics 37, no. 2 (June 30, 2006): 135–48. http://dx.doi.org/10.5556/j.tkjm.37.2006.158.

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The aim of this paper is to derive corrected Bullen-Simpson's 3/8 inequality, starting from corrected Simpson's 3/8 and corrected Maclaurin's formula. By corrected we mean formulae that approximate the integral not only with the values of the function in certain points but also with the value of the first derivative in end points of the interval. These formulae will have a higher degree of exactness than formulae derived in [3].

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9

Borwein, Jonathan M., David Borwein, and William F. Galway. "Finding and Excluding b-ary Machin-Type Individual Digit Formulae." Canadian Journal of Mathematics 56, no. 5 (October 1, 2004): 897–925. http://dx.doi.org/10.4153/cjm-2004-041-2.

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AbstractConstants with formulae of the form treated by D. Bailey, P. Borwein, and S. Plouffe (BBP formulae to a given base b) have interesting computational properties, such as allowing single digits in their base b expansion to be independently computed, and there are hints that they should be normal numbers, i.e., that their base b digits are randomly distributed. We study a formally limited subset of BBP formulae, which we call Machin-type BBP formulae, for which it is relatively easy to determine whether or not a given constant κ has a Machin-type BBP formula. In particular, given b ∈ ℕ, b > 2, b not a proper power, a b-ary Machin-type BBP arctangent formula for κ is a formula of the form κ = Σmam arctan(–b–m), am ∈ ℚ, while when b = 2, we also allow terms of the form am arctan(1/(1 – 2m)). Of particular interest, we show that π has no Machin-type BBP arctangent formula when b ≠ 2. To the best of our knowledge, when there is no Machin-type BBP formula for a constant then no BBP formula of any form is known for that constant.

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10

Dedić, Lj, M. Matić, and Josip Pečarić. "Euler-Maclaurin formulae." Mathematical Inequalities & Applications, no. 2 (2003): 247–75. http://dx.doi.org/10.7153/mia-06-24.

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11

Yiu, Paul Y. H. "Sums of Squares Formulae With Integer Coefficients." Canadian Mathematical Bulletin 30, no. 3 (September 1, 1987): 318–24. http://dx.doi.org/10.4153/cmb-1987-045-6.

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AbstractHidden behind a sums of squares formula are other such formulae not obtainable by restriction. This drastically simplifies the combinatorics involved in the existence problem of sums of squares formulae, and leads to a proof that the product of two sums of 16 squares cannot be rewritten as a sum of 28 squares, if only integer coefficients are permitted. We also construct all [10, 10, 16] formulae.

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12

Soni, R. C., and Deepika Singh. "The unified Riemann-Liouville fractional derivative formulae." Tamkang Journal of Mathematics 36, no. 3 (September 30, 2005): 231–36. http://dx.doi.org/10.5556/j.tkjm.36.2005.115.

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In this paper, we obtain two unified fractional derivative formulae. The first involves the product of two general class of polynomials and the multivariable $H$-function. The second fractional derivative formula also involves the product of two general class of polynomials and the multivariable $H$-function and has been obtained by the application of the first fractional derivative formula twice and it has two independent variables instead of one. The polynomials and the functions involved in both the fractional derivative formulae as well as their arguments are quite general in nature and so our findings provide interesting unifications and extensions of a number of (known and new) results. For the sake of illustration, we point out that the fractional derivative formulae recently obtained by Srivastava, Chandel and Vishwakarma [11], Srivastava and Goyal [12], Gupta, Agrawal and Soni [4], Gupta and Agrawal [3] follow as particular cases of our findings. In the end, we record a new fractional derivative formula involving the product of the Konhauser biorthogonal polynomials, the Jacobi polynomials and the product of $r$ different modified Bessel functions of the second kind as a simple special case of our first formula.

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13

AGLIĆ ALJINOVIĆ, A., J. PEČARIĆ, and M. RIBIČIĆ PENAVA. "SHARP INTEGRAL INEQUALITIES BASED ON GENERAL TWO-POINT FORMULAE VIA AN EXTENSION OF MONTGOMERY’S IDENTITY." ANZIAM Journal 51, no. 1 (July 2009): 67–101. http://dx.doi.org/10.1017/s1446181109000315.

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AbstractWe consider families of general two-point quadrature formulae, using the extension of Montgomery’s identity via Taylor’s formula. The formulae obtained are used to present a number of inequalities for functions whose derivatives are fromLpspaces and Bullen-type inequalities.

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14

Ahn, Jaehyun, Soyoung Choi, and Hwanyup Jung. "Class number formulae in the form of a product of determinants in function fields." Journal of the Australian Mathematical Society 78, no. 2 (April 2005): 227–38. http://dx.doi.org/10.1017/s1446788700008053.

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AbstractIn this paper, we generalize the Kučera's group-determinant formulae to obtain the real and relative class number formulae of any subfield of cyclotomic function fields with arbitrary conductor in the form of a product of determinants. From these formulae, we generalize the relative class number formula of Rosen and Bae-Kang and obtain analogous results of Tsumura and Hirabayashi for an intermediate field in the tower of cyclotomic function fields with prime power conductor.

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15

Dedić, LJ, M. Matić, and J. Pečarić. "On Euler midpoint formulae." ANZIAM Journal 46, no. 3 (January 2005): 417–38. http://dx.doi.org/10.1017/s144618110000835x.

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AbstractModified versions of the Euler midpoint formula are given for functions whose derivatives are either functions of bounded variation, Lipschitzian functions or functions in Lp-spaces. The results are applied to quadrature formulae.

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16

Boyanov, B. D. "Optimal quadrature formulae." Russian Mathematical Surveys 60, no. 6 (December 31, 2005): 1035–55. http://dx.doi.org/10.1070/rm2005v060n06abeh004280.

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17

Dangerfield, Janet. "Formulae for Advanced Mathematics with Statistical Tables (SMP)." Mathematical Gazette 69, no. 448 (June 1985): 145. http://dx.doi.org/10.2307/3616950.

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18

Steven, Zelditch. "Kuznecov Sum Formulae and Szegˇ Limit Formulae on Manifolds." Communications in Partial Differential Equations 17, no. 1-2 (January 1992): 221–60. http://dx.doi.org/10.1080/03605309208820840.

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19

Morikawa, Tetsuo. "Graph‐theoretical identification of molecular formulae, empirical formulae, and molecular fragments." International Journal of Mathematical Education in Science and Technology 18, no. 4 (July 1987): 555–60. http://dx.doi.org/10.1080/0020739870180407.

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20

Chernyshenko, V. M., and Ye I. Riabchenko. "On tangent iteration formulae." Researches in Mathematics, no. 2 (July 10, 2021): 66. http://dx.doi.org/10.15421/246916.

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21

Dedić, Lj, M. Matić, and J. Pečarić. "On Euler trapezoid formulae." Applied Mathematics and Computation 123, no. 1 (September 2001): 37–62. http://dx.doi.org/10.1016/s0096-3003(00)00054-0.

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22

Gushev, Vesselin, and Geno Nikolov. "Modified product cubature formulae." Journal of Computational and Applied Mathematics 224, no. 2 (February 2009): 465–75. http://dx.doi.org/10.1016/j.cam.2008.05.031.

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23

Pečarić, Josip, Ivan Perić, and Ana Vukelić. "On Euler-Boole formulae." Mathematical Inequalities & Applications, no. 1 (2004): 27–46. http://dx.doi.org/10.7153/mia-07-05.

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24

Fiedler, Miroslav. "Remarks on the Sherman-Morrison-Woodbury formulae." Mathematica Bohemica 128, no. 3 (2003): 253–62. http://dx.doi.org/10.21136/mb.2003.134181.

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25

Teichmann, Josef. "Calculating the Greeks by cubature formulae." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 462, no. 2066 (December 14, 2005): 647–70. http://dx.doi.org/10.1098/rspa.2005.1583.

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We provide cubature formulae for the calculation of derivatives of expected values in the spirit of Terry Lyons and Nicolas Victoir. In financial mathematics derivatives of option prices with respect to initial values, so called Greeks, are of particular importance as hedging parameters. The proof of existence of cubature formulae for Greeks is based on universal formulae, which lead to the calculation of Greeks in an asymptotic sense—even without Hörmander's condition. Cubature formulae then allow to calculate these quantities very quickly. Simple examples are added to the theoretical exposition.

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26

Moore, Stephen T. "Diagrammatic morphisms between indecomposable modules of Ūq(𝔰𝔩2)." International Journal of Mathematics 31, no. 02 (January 28, 2020): 2050016. http://dx.doi.org/10.1142/s0129167x20500160.

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We give diagrammatic formulae for morphisms between indecomposable representations of [Formula: see text] appearing in the decomposition of [Formula: see text], including projections and second endomorphisms on projective indecomposable representations.

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27

Frontczak, Maria, and Andrzej Miodek. "Weil's formulae and multiplicity." Annales Polonici Mathematici 55, no. 1 (1991): 103–8. http://dx.doi.org/10.4064/ap-55-1-103-108.

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28

Kraus, Jon, and David R. Larson. "Reflexivity and Distance Formulae." Proceedings of the London Mathematical Society s3-53, no. 2 (September 1986): 340–56. http://dx.doi.org/10.1112/plms/s3-53.2.340.

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29

Akhiezer, Dmitri, and Boris Kazarnovskii. "Crofton Formulae for Products." Moscow Mathematical Journal 22, no. 3 (2022): 377–92. http://dx.doi.org/10.17323/1609-4514-2022-22-3-377-392.

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30

Franjić, I., and J. Pečarić. "Corrected Euler-Maclaurin’s formulae." Rendiconti del Circolo Matematico di Palermo 54, no. 2 (June 2005): 259–72. http://dx.doi.org/10.1007/bf02874640.

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31

Berens, H., H. J. Schmid, and Y. Xu. "Multivariate Gaussian cubature formulae." Archiv der Mathematik 64, no. 1 (January 1995): 26–32. http://dx.doi.org/10.1007/bf01193547.

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32

Meunier, Frédéric. "Combinatorial Stokes formulae." European Journal of Combinatorics 29, no. 1 (January 2008): 286–97. http://dx.doi.org/10.1016/j.ejc.2006.07.010.

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33

Zặlinescu, C. "A comparison of constraint qualifications in infinite-dimensional convex programming revisited." Journal of the Australian Mathematical Society. Series B. Applied Mathematics 40, no. 3 (January 1999): 353–78. http://dx.doi.org/10.1017/s033427000001095x.

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In 1990 Gowda and Teboulle published the paper [16], making a comparison of several conditions ensuring the Fenchel-Rockafellar duality formulainf{f(x) + g(Ax) | x ∈ X} = max{−f*(A*y*) − g*(− y*) | y* ∈ Y*}.Probably the first comparison of different constraint qualification conditions was made by Hiriart-Urruty [17] in connection with ε-subdifferential calculus. Among them appears, as the basic sufficient condition, the formula for the conjugate of the corresponding function; such functions are: f1 + f2, g o A, max{fl,…, fn}, etc. In fact strong duality formulae (like the one above) and good formulae for conjugates are equivalent and they can be used to obtain formulae for ε-subdifferentials, using a technique developed in [17] and extensively used in [46].

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34

Caminati, Marco B., and Artur Korniłowicz. "Pseudo-Canonical Formulae are Classical." Formalized Mathematics 22, no. 2 (June 30, 2014): 99–103. http://dx.doi.org/10.2478/forma-2014-0011.

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Summary An original result about Hilbert Positive Propositional Calculus introduced in [11] is proven. That is, it is shown that the pseudo-canonical formulae of that calculus (and hence also the canonical ones, see [17]) are a subset of the classical tautologies.

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35

Schmid, H. J., and Yuan Xu. "On bivariate Gaussian cubature formulae." Proceedings of the American Mathematical Society 122, no. 3 (March 1, 1994): 833. http://dx.doi.org/10.1090/s0002-9939-1994-1209428-0.

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36

Cools, R., I. P. Mysovskikh, and H. J. Schmid. "Cubature formulae and orthogonal polynomials." Journal of Computational and Applied Mathematics 127, no. 1-2 (January 2001): 121–52. http://dx.doi.org/10.1016/s0377-0427(00)00495-7.

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37

Wang, Xiaoqun. "On generalized invariant cubature formulae." Journal of Computational and Applied Mathematics 130, no. 1-2 (May 2001): 271–81. http://dx.doi.org/10.1016/s0377-0427(99)00377-5.

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38

Degani, Ilan, Jeremy Schiff, and David J. Tannor. "Commuting extensions and cubature formulae." Numerische Mathematik 101, no. 3 (July 18, 2005): 479–500. http://dx.doi.org/10.1007/s00211-005-0628-z.

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39

Catarino, Paula. "k-Pell, k-Pell–Lucas and modified k-Pell sedenions." Asian-European Journal of Mathematics 12, no. 02 (April 2019): 1950018. http://dx.doi.org/10.1142/s1793557119500189.

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The aim of this work is to present the [Formula: see text]-Pell, the [Formula: see text]-Pell–Lucas and the Modified [Formula: see text]-Pell sedenions and we give some properties involving these sequences, including the Binet-style formulae and the ordinary generating functions.

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40

Lanzara, Flavia. "On optimal quadrature formulae." Journal of Inequalities and Applications 2000, no. 3 (2000): 923754. http://dx.doi.org/10.1155/s1025583400000114.

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41

Kalorkoti, K. "Types of depth and formula size." Asian-European Journal of Mathematics 07, no. 02 (June 2014): 1450031. http://dx.doi.org/10.1142/s1793557114500314.

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We use a rank-based measure on rational expressions in indeterminates over a field and define notions of size and depth with associated subparts of formulae for expressions. Formulae are allowed to have as inputs expressions from a large set rather than just constants and indeterminates. A general lower bound is derived and this is used to deduce an exponential lower bound, subject to depth assumptions, on the formula size of the determinant with inputs restricted to the usual constants and indeterminates. The general bound is also used to show that a polynomial which is closely related to the determinant has exponential formula size if either (i) some types of operations do not occur in the formula or (ii) some assumptions on depth hold (the inputs allowed here are from a large set).

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42

Masjed-Jamei, Mohammad, Gradimir Milovanovic, and M. A. Jafari. "Closed expressions for coefficients in weighted Newton-Cotes quadratures." Filomat 27, no. 4 (2013): 649–58. http://dx.doi.org/10.2298/fil1304649m.

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In this short note, we derive closed expressions for Cotes numbers in the weighted Newton-Cotes quadrature formulae with equidistant nodes in terms of moments and Stirling numbers of the first kind. Three types of equidistant nodes are considered. The corresponding program codes in Mathematica Package are presented. Finally, in order to illustrate the application of the obtained quadrature formulas a few numerical examples are included.

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43

Alam, Noor, Waseem Ahmad Khan, Can Kızılateş, Sofian Obeidat, Cheon Seoung Ryoo, and Nabawia Shaban Diab. "Some Explicit Properties of Frobenius–Euler–Genocchi Polynomials with Applications in Computer Modeling." Symmetry 15, no. 7 (July 4, 2023): 1358. http://dx.doi.org/10.3390/sym15071358.

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Many properties of special polynomials, such as recurrence relations, sum formulas, and symmetric properties, have been studied in the literature with the help of generating functions and their functional equations. In this study, we define Frobenius–Euler–Genocchi polynomials and investigate some properties by giving many relations and implementations. We first obtain different relations and formulas covering addition formulas, recurrence rules, implicit summation formulas, and relations with the earlier polynomials in the literature. With the help of their generating function, we obtain some new relations, including the Stirling numbers of the first and second kinds. We also obtain some new identities and properties of this type of polynomial. Moreover, using the Faà di Bruno formula and some properties of the Bell polynomials of the second kind, we obtain an explicit formula for the Frobenius–Euler polynomials of order α. We provide determinantal representations for the ratio of two differentiable functions. We find a recursive relation for the Frobenius–Euler polynomials of order α. Using the Mathematica program, the computational formulae and graphical representation for the aforementioned polynomials are obtained.

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44

Brigo, Damiano, Cristin Buescu, Marco Francischello, Andrea Pallavicini, and Marek Rutkowski. "Nonlinear Valuation with XVAs: Two Converging Approaches." Mathematics 10, no. 5 (March 2, 2022): 791. http://dx.doi.org/10.3390/math10050791.

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When pricing OTC contracts in the presence of additional risk factors and costs, such as credit risk and funding and collateral costs, the starting “clean price” is modified additively by valuation adjustments (XVAs) that account for each factor or cost in isolation, while seemingly ignoring the combined effects. Instead, risk factors and costs can be jointly accounted for ab initio in the pricing mechanism at the level of cash flows, and this “adjusted cash flow" approach leads to a nonlinear valuation formula. While for practitioners this made more sense because it showed which discount factor is used for which cash flow (recall the multi-curve environment post-crisis), for academics, the focus was on checking that the resulting nonlinear valuation formula is consistent with the theoretical arbitrage-free “replication approach” that we also analyse in the paper. We formulate specific reasonable assumptions, which ensure that the valuation formulae obtained by the two approaches coincide, thus reinforcing both academics’ and practitioners’ confidence in adopting such nonlinear valuation formulae in a multi-curve setup.

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45

Yang, Kichoon. "Plücker formulae for the orthogonal group." Bulletin of the Australian Mathematical Society 40, no. 3 (December 1989): 447–56. http://dx.doi.org/10.1017/s0004972700017512.

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Plücker formulae for horizontal curves in SO(m)-flag manifolds are derived. These formulae are seen to generalise the usual Plücker formulae for projective space curves. They also have applications in the theory of minimal surfaces in Euclidean sphere and the complex hyperquadric.

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46

Jandrlic, Davorka, Miodrag Spalevic, and Jelena Tomanovic. "Error estimates for certain cubature formulae." Filomat 32, no. 20 (2018): 6893–902. http://dx.doi.org/10.2298/fil1820893j.

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We estimate the errors of selected cubature formulae constructed by the product of Gauss quadrature rules. The cases of multiple and (hyper-)surface integrals over n-dimensional cube, simplex, sphere and ball are considered. The error estimates are obtained as the absolute value of the difference between cubature formula constructed by the product of Gauss quadrature rules and cubature formula constructed by the product of corresponding Gauss-Kronrod or corresponding generalized averaged Gaussian quadrature rules. Generalized averaged Gaussian quadrature rule ?2l+1 is (2l + 1)-point quadrature formula. It has 2l + 1 nodes and the nodes of the corresponding Gauss rule Gl with l nodes form a subset, similar to the situation for the (2l + 1)-point Gauss-Kronrod rule H2l+1 associated with Gl. The advantages of bG2l+1 are that it exists also when H2l+1 does not, and that the numerical construction of ?2l+1, based on recently proposed effective numerical procedure, is simpler than the construction of H2l+1.

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47

IKEDA, YASUWO, KOHJI TOMITA, and CHIHARU HOSONO. "On the elimination of some higher type quantifiers." Mathematical Structures in Computer Science 11, no. 6 (December 2001): 771–79. http://dx.doi.org/10.1017/s0960129501003401.

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This paper concerns the elimination of higher type quantifiers and gives two theorems. The first theorem shows that quantifiers in formulae of a specific form can be eliminated. The second theorem shows that quantifiers in formulae of a similar form cannot be eliminated, that is, such formulae do not have an equivalent first-order formula. The proof is based on the Ehrenfeucht game. These theorems are important for design of an interpreter of a ν act, which is a representation of mathematical action. Moreover, even if the universe is assumed to be finite, these theorems hold.

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Frenkel, Péter E. "Character formulae for classical groups." Central European Journal of Mathematics 4, no. 2 (June 2006): 242–49. http://dx.doi.org/10.2478/s11533-006-0004-y.

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Dedić, Lj, M. Matić, and J. Pečarić. "On dual Euler-Simpson formulae." Bulletin of the Belgian Mathematical Society - Simon Stevin 8, no. 3 (2001): 479–504. http://dx.doi.org/10.36045/bbms/1102714571.

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KUMAGAI, Hiroshi. "ON UNIFIED KRONECKER LIMIT FORMULAE." Kyushu Journal of Mathematics 56, no. 1 (2002): 41–51. http://dx.doi.org/10.2206/kyushujm.56.41.

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