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Journal articles on the topic 'Quadratic Formula'

Author: Grafiati

Published: 4 June 2021

Last updated: 25 July 2025

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1

March, Robert H. "The mystical ‘‘quadratic formula’’." Physics Teacher 31, no. 3 (1993): 147. http://dx.doi.org/10.1119/1.2343692.

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2

Hungerbühler, Norbert. "An alternative quadratic formula." Mathematische Semesterberichte 67, no. 1 (2019): 85–95. http://dx.doi.org/10.1007/s00591-019-00262-3.

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3

Karlsson, Per, and Thomas Ernst. "Applications of quadratic and cubic hypergeometric transformations." Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica 78, no. 1 (2024): 37–73. https://doi.org/10.17951/a.2024.78.1.37-73.

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The purpose of this paper is to consider five classes of quadratic and cubic hypergeometric transformations in the spirit of Bailey and Whipple. We shall successfully evaluate several hypergeometric functions, of the types 2F1(x), 3F2(x), and 4F3(x), with each function having one or more free parameters, and with the argument $x$ chosen to equal such unusual values as x=±1,−8,14,−18, (these four values having been explored initially by Gessel and Stanton). In each case, companion identities and/or inverse transformations are given, which are sometimes proved by a limiting process for a diverge

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4

Jones, Edna. "Local densities of diagonal integral ternary quadratic forms at odd primes." International Journal of Number Theory 17, no. 03 (2021): 547–75. http://dx.doi.org/10.1142/s1793042120400357.

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We give formulas for local densities of diagonal integral ternary quadratic forms at odd primes. Exponential sums and quadratic Gauss sums are used to obtain these formulas. These formulas (along with 2-adic densities and Siegel’s mass formula) can be used to compute the representation numbers of certain ternary quadratic forms.

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5

ALACA, AYŞE, ŞABAN ALACA, and KENNETH S. WILLIAMS. "SEXTENARY QUADRATIC FORMS AND AN IDENTITY OF KLEIN AND FRICKE." International Journal of Number Theory 06, no. 01 (2010): 169–83. http://dx.doi.org/10.1142/s1793042110002880.

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Formulae, originally conjectured by Liouville, are proved for the number of representations of a positive integer n by each of the eight sextenary quadratic forms [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text].

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6

Wen, Hui, and Feng Ling Li. "A Simplified Formula to Calculate the Initial Value of Iteration for Contracted Depth in Quadratic Parabola Shaped Channels." Applied Mechanics and Materials 744-746 (March 2015): 1039–44. http://dx.doi.org/10.4028/www.scientific.net/amm.744-746.1039.

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At present, the complexity of calculation process and expression form of the initial value of iteration for contracted depth in quadratic parabola shaped channels,Seek a new iterative initial value formula for contracted depth in quadratic parabola shaped channels. Through an identical deformation on the basic equation for contracted depth in quadratic parabola shaped channels. Deduce the iterative formula for computing the quadratic parabola section contraction water depth. Introduction the dimensionless contraction water depth concept, plot the dimensionless contraction water depth and the d

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7

Alaca, Ayşe, and M. Nesibe Kesicioğlu. "Representations by octonary quadratic forms with coefficients 1, 2, 3 or 6." International Journal of Number Theory 13, no. 03 (2017): 735–49. http://dx.doi.org/10.1142/s1793042117500385.

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Using modular forms, we determine formulas for the number of representations of a positive integer by diagonal octonary quadratic forms with coefficients [Formula: see text], [Formula: see text], [Formula: see text] or [Formula: see text].

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8

Ramakrishnan, B., Brundaban Sahu, and Anup Kumar Singh. "On the number of representations by certain octonary quadratic forms with coefficients 1, 2, 3, 4 and 6." International Journal of Number Theory 14, no. 03 (2018): 751–812. http://dx.doi.org/10.1142/s1793042118500483.

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In this paper, we find formulas for the number of representations of certain diagonal octonary quadratic forms with coefficients [Formula: see text] and [Formula: see text]. We obtain these formulas by constructing explicit bases of the space of modular forms of weight [Formula: see text] on [Formula: see text] with character.

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9

Clarke, Robert J. "82.48 The Quadratic Equation Formula." Mathematical Gazette 82, no. 495 (1998): 460. http://dx.doi.org/10.2307/3619896.

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10

Mahajan, Sanjoy. "Fleeing from the quadratic formula." American Journal of Physics 87, no. 5 (2019): 332–34. http://dx.doi.org/10.1119/1.5097757.

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11

Lomadze, G. "On the Number of Representations of Positive Integers by the Quadratic Form." gmj 8, no. 1 (2001): 111–27. http://dx.doi.org/10.1515/gmj.2001.111.

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Abstract An explicit exact (non asymptotic) formula is derived for the number of representations of positive integers by the quadratic form . The way by which this formula is derived, gives us a possibility to develop a method of finding the so-called Liouville type formulas for the number of representations of positive integers by positive diagonal quadratic forms in nine variables with integral coefficients.

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12

0jo, O. A., and A. M. Gbolagade. "COMPARISON BETWEEN THE USE OF QUADRATIC FORMULA AND COMPLETING THE SQUARE METHODS IN SOLVING QUADRATIC EQUATIONS BY SENIOR SECONDARY STUDENTS." Nigerian Educational Digest (NED) Volume 12, No. 1, June 2012 (2012): 97–104. https://doi.org/10.5281/zenodo.7775461.

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The study examined the use of the “quadratic formula method” and “completing the square method” in solving quadratic equations on Senior Secondary students’ Mathematics. It also determined the differential effects between boys and girls when solving quadratic equation by using the methods of quadratic formula and completing the square. Data were collected from 200 S.S.2 students using Quadratic Equation Mathematics Achievement Test (QEMAT). Two null hypotheses were generated for the study. Results from the study revealed that students taught with quadratic formula

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13

Bugeaud, Yann, та Tomislav Pejković. "Quadratic approximation in ℚp". International Journal of Number Theory 11, № 01 (2014): 193–209. http://dx.doi.org/10.1142/s1793042115500128.

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Let p be a prime number. Let w2 and [Formula: see text] denote the exponents of approximation defined by Mahler and Koksma, respectively, in their classifications of p-adic numbers. It is well-known that every p-adic number ξ satisfies [Formula: see text], with [Formula: see text] for almost all ξ. By means of Schneider's continued fractions, we give explicit examples of p-adic numbers ξ for which the function [Formula: see text] takes any prescribed value in the interval (0, 1].

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14

Banks, William D., and Victor Z. Guo. "Quadratic nonresidues below the Burgess bound." International Journal of Number Theory 13, no. 03 (2017): 751–59. http://dx.doi.org/10.1142/s1793042117500397.

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For any odd prime number [Formula: see text], let [Formula: see text] be the Legendre symbol, and let [Formula: see text] be the sequence of positive nonresidues modulo [Formula: see text], i.e. [Formula: see text] for each [Formula: see text]. In 1957, Burgess showed that the upper bound [Formula: see text] holds for any fixed [Formula: see text]. In this paper, we prove that the stronger bound [Formula: see text] holds for all odd primes [Formula: see text] provided that [Formula: see text] where the implied constants are absolute. For fixed [Formula: see text], we also show that there is a

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15

KÖKLÜCE, BÜLENT. "REPRESENTATION NUMBERS OF TWO OCTONARY QUADRATIC FORMS." International Journal of Number Theory 09, no. 07 (2013): 1641–48. http://dx.doi.org/10.1142/s1793042113500486.

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Let N(a1, …, a4; n) denote the number of representations of an integer n by the form [Formula: see text]. In this paper we derive formulae for N(1, 1, 1, 2; n) and N(1, 2, 2, 2; n). These formulae are given in terms of σ3(n).

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16

Tian, Rushun, Zhi-Qiang Wang, and Leiga Zhao. "Schrödinger systems with quadratic interactions." Communications in Contemporary Mathematics 21, no. 08 (2019): 1850077. http://dx.doi.org/10.1142/s0219199718500773.

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In this paper, we consider the existence and multiplicity of nontrivial solutions to a quadratically coupled Schrödinger system [Formula: see text] where [Formula: see text], [Formula: see text] and [Formula: see text] are constants and [Formula: see text], [Formula: see text]. Such type of systems stem from applications in nonlinear optics, Bose–Einstein condensates and plasma physics. The existence (and nonexistence), multiplicity and asymptotic behavior of vector solutions of the system are established via variational methods. In particular, for multiplicity results we develop new technique

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17

Ma, Shouhei. "Equivariant Gauss sum of finite quadratic forms." Forum Mathematicum 30, no. 4 (2018): 1029–47. http://dx.doi.org/10.1515/forum-2017-0070.

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Abstract The classical quadratic Gauss sum can be thought of as an exponential sum attached to a quadratic form on a cyclic group. We introduce an equivariant version of Gauss sum for arbitrary finite quadratic forms, which is an exponential sum twisted by the action of the orthogonal group. We prove that simple arithmetic formulas hold for some basic classes of quadratic forms. In applications, such invariant appears in the dimension formula for certain vector-valued modular forms.

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18

Vlasenko, Maria, and Don Zagier. "Higher Kronecker “limit” formulas for real quadratic fields." Journal für die reine und angewandte Mathematik (Crelles Journal) 2013, no. 679 (2013): 23–64. http://dx.doi.org/10.1515/crelle.2012.022.

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Abstract For every integer k ≧ 2 we introduce an analytic function of a positive real variable and give a universal formula expressing the values ζ(ℬ, k) of the zeta functions of narrow ideal classes in real quadratic fields in terms of this function and its derivatives up to order k − 1 evaluated at reduced real quadratic irrationalities associated to ℬ. We show that our functions satisfy functional equations and use these to deduce explicit formulas for the rational numbers ζ(ℬ, 1 − k). We also give an interpretation of our formula for ζ(ℬ, k) in terms of cohomology groups of SL(2, ℤ) with a

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19

Taki Eldin, Ramy F. "On the number of incongruent solutions to a quadratic congruence over algebraic integers." International Journal of Number Theory 15, no. 01 (2019): 105–30. http://dx.doi.org/10.1142/s1793042118501762.

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Over the ring of algebraic integers [Formula: see text] of a number field [Formula: see text], the quadratic congruence [Formula: see text] modulo a nonzero ideal [Formula: see text] is considered. We prove explicit formulas for [Formula: see text] and [Formula: see text], the number of incongruent solutions [Formula: see text] and the number of incongruent solutions [Formula: see text] with [Formula: see text] coprime to [Formula: see text], respectively. If [Formula: see text] is contained in a prime ideal [Formula: see text] containing the rational prime [Formula: see text], it is assumed t

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20

Rababah, Abedallah. "Quadratic quadrature formula for curves with third degree of exactness." Filomat 33, no. 2 (2019): 457–62. http://dx.doi.org/10.2298/fil1902457r.

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In this article, a quadrature formula of degree 2 is given that has degree of exactness 3 and order 5. The formula is valid for any planar curve given in parametric form unlike existing Gaussian quadrature formulas that are valid only for functions.

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21

Frank, Kristin. "The Structure of the Quadratic Formula." Mathematics Teacher: Learning and Teaching PK-12 114, no. 5 (2021): 395–98. http://dx.doi.org/10.5951/mtlt.2020.0193.

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22

Sastry, K. R. S. "The Quadratic Formula: A Historic Approach." Mathematics Teacher 81, no. 8 (1988): 670–72. http://dx.doi.org/10.5951/mt.81.8.0670.

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23

Edwards, Thomas G., and Kenneth R. Chelst. "Finding Meaning in the Quadratic Formula." Mathematics Teacher 112, no. 4 (2019): 258–61. http://dx.doi.org/10.5951/mathteacher.112.4.0258.

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24

Sander, J. W. "A reciprocity formula for quadratic forms." Monatshefte f�r Mathematik 104, no. 2 (1987): 125–32. http://dx.doi.org/10.1007/bf01326785.

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25

KÖKLÜCE, BÜLENT. "THE REPRESENTATION NUMBERS OF CERTAIN OCTONARY QUADRATIC FORMS." International Journal of Number Theory 09, no. 05 (2013): 1125–39. http://dx.doi.org/10.1142/s1793042113500164.

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26

Wang, Song, and Linsheng Zhu. "Non-degenerate Invariant Bilinear Forms on Leibniz Algebras." Algebra Colloquium 22, no. 04 (2015): 711–20. http://dx.doi.org/10.1142/s1005386715000607.

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In this paper, we study Leibniz algebras [Formula: see text] with a non-degenerate Leibniz-symmetric [Formula: see text]-invariant bilinear form B, such a pair [Formula: see text] is called a quadratic Leibniz algebra. Our first result generalizes the notion of double extensions to quadratic Leibniz algebras. This notion was introduced by Medina and Revoy to study quadratic Lie algebras. In the second theorem, we give a sufficient condition for a quadratic Leibniz algebra to be a quadratic Leibniz algebra by double extension.

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27

Patane, Frank. "A proof of Hecke’s formula for binary quadratic forms." International Journal of Number Theory 16, no. 02 (2019): 233–40. http://dx.doi.org/10.1142/s179304212050013x.

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In Mathematische Werke, Hecke defines the operator [Formula: see text] and describes their utility in conjunction with theta series of quadratic forms. In particular, he shows that the image of theta series associated to classes of binary quadratic forms in CL[Formula: see text] is again a theta series associated to a collection of forms in CL[Formula: see text]. We state and prove an explicit formula for the action of [Formula: see text] on a binary quadratic form of negative discriminant.

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28

HAYASHI, NAKAO, and PAVEL I. NAUMKIN. "SUBCRITICAL QUADRATIC NONLINEAR SCHRÖDINGER EQUATION." Communications in Contemporary Mathematics 13, no. 06 (2011): 969–1007. http://dx.doi.org/10.1142/s021919971100452x.

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We study the initial value problem for the quadratic nonlinear Schrödinger equation [Formula: see text] where γ > 0. Suppose that the Fourier transform [Formula: see text] of the initial data u1satisfies estimates [Formula: see text], where ε > 0 is sufficiently small. Also suppose that [Formula: see text] for |ξ| ≤ 1. Assume that γ > 0 is small: [Formula: see text]. Then we prove that there exists a unique solution u ∈ C([1, ∞);L2) of the Cauchy problem (*). Moreover, the solution u approaches for large time t → +∞ a self-similar solution of the quadratic nonlinear Schrödinger equati

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29

Sze, Sandra. "Explicit solutions of imaginary quadratic norm equations." International Journal of Number Theory 15, no. 08 (2019): 1635–73. http://dx.doi.org/10.1142/s1793042119500921.

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Let [Formula: see text] be an imaginary quadratic extension of [Formula: see text]. Let [Formula: see text] be the class number and [Formula: see text] be the discriminant of the field [Formula: see text]. Assume [Formula: see text] is a prime such that [Formula: see text]. Then [Formula: see text] splits in [Formula: see text]. The elements of the ring of integers [Formula: see text] are of the form [Formula: see text] if [Formula: see text] and [Formula: see text] if [Formula: see text], where [Formula: see text] and [Formula: see text]. The norm [Formula: see text] and [Formula: see text],

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30

Milovic, Djordjo Z. "On the 8-rank of narrow class groups of ℚ(−4pq), ℚ(−8pq), and ℚ(8pq)". International Journal of Number Theory 14, № 08 (2018): 2165–93. http://dx.doi.org/10.1142/s1793042118501300.

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Let [Formula: see text]. We study the [Formula: see text]-part of the narrow class group in thin families of quadratic number fields of the form [Formula: see text], where [Formula: see text] are prime numbers, and we prove new lower bounds for the proportion of narrow class groups in these families that have an element of order [Formula: see text]. In the course of our proof, we prove a general double-oscillation estimate for the quadratic residue symbol in quadratic number fields.

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31

Zhou, Jinsen, and Yanyong Hong. "Quadratic Leibniz conformal algebras." Journal of Algebra and Its Applications 18, no. 10 (2019): 1950195. http://dx.doi.org/10.1142/s0219498819501950.

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In this paper, we study a class of Leibniz conformal algebras called quadratic Leibniz conformal algebras. An equivalent characterization of a Leibniz conformal algebra [Formula: see text] through three algebraic operations on [Formula: see text] is given. By this characterization, several constructions of quadratic Leibniz conformal algebras are presented. Moreover, one-dimensional central extensions of quadratic Leibniz conformal algebras are considered using some bilinear forms on [Formula: see text]. In particular, we also study one-dimensional Leibniz central extensions of quadratic Lie c

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32

DeMarco, Laura, Holly Krieger, and Hexi Ye. "Common preperiodic points for quadratic polynomials." Journal of Modern Dynamics 18 (2022): 363. http://dx.doi.org/10.3934/jmd.2022012.

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<p style='text-indent:20px;'>Let <inline-formula><tex-math id="M1">\begin{document}$ f_c(z) = z^2+c $\end{document}</tex-math></inline-formula> for <inline-formula><tex-math id="M2">\begin{document}$ c \in {\mathbb C} $\end{document}</tex-math></inline-formula>. We show there exists a uniform upper bound on the number of points in <inline-formula><tex-math id="M3">\begin{document}$ {\mathbb P}^1( {\mathbb C}) $\end{document}</tex-math></inline-formula> that can be preperiodic for both <inline-formula><tex-ma

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33

XIA, ERNEST X. W., and OLIVIA X. M. YAO. "ON THE REPRESENTATIONS OF INTEGERS BY CERTAIN QUADRATIC FORMS." International Journal of Number Theory 09, no. 01 (2012): 189–204. http://dx.doi.org/10.1142/s1793042112501333.

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In this paper, using the (p, k)-parametrization of theta functions given by Alaca, Alaca and Williams, we establish some theta function identities. Explicit formulas are obtained for the number of representations of a positive integer n by the quadratic forms [Formula: see text] with a ≠ 0, a + b + c = 4 and [Formula: see text] with k + l = 2 and r + s + t = 2 by employing these identities.

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34

Hu, Ying, Remi Moreau, and Falei Wang. "Quadratic mean-field reflected BSDEs." Probability, Uncertainty and Quantitative Risk 7, no. 3 (2022): 169. http://dx.doi.org/10.3934/puqr.2022012.

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<p style='text-indent:20px;'>In this paper, we analyze mean-field reflected backward stochastic differential equations when the driver has quadratic growth in the second unknown <inline-formula><tex-math id="M1">\begin{document}$ z $\end{document}</tex-math></inline-formula>. Using a linearization technique and the BMO martingale theory, we first apply a fixed-point argument to establish the uniqueness and existence result for the case with bounded terminal condition and obstacle. Then, with the help of the <inline-formula><tex-math id="M2">\begin{docu

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35

Sun, Zhi-Wei. "Quadratic residues and quartic residues modulo primes." International Journal of Number Theory 16, no. 08 (2020): 1833–58. http://dx.doi.org/10.1142/s1793042120500955.

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In this paper, we study some products related to quadratic residues and quartic residues modulo primes. Let [Formula: see text] be an odd prime and let [Formula: see text] be any integer. We determine completely the product [Formula: see text] modulo [Formula: see text]; for example, if [Formula: see text] then [Formula: see text] where [Formula: see text] denotes the Legendre symbol. We also determine [Formula: see text] modulo [Formula: see text].

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36

Nassrallah, Bassam. "Basic Double Series, Quadratic Transformations and Products of Basic Series." Canadian Mathematical Bulletin 34, no. 4 (1991): 499–513. http://dx.doi.org/10.4153/cmb-1991-080-7.

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AbstractA basic double series is expressed in terms of two 5ϕ4 series which extends Bailey's transformation of an 8ϕ7 series into two 4ϕ3 's. From this formula we derive some quadratic transformations; one of them is a new q-analogue of a transformation due to Whipple. Product formulas as well as Gasper-Rahman's q-Clausen formula are also given as special cases.

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37

Llibre, Jaume, Arefeh Nabavi, and Marzieh Mousavi. "Limit Cycles Bifurcating from a Family of Reversible Quadratic Centers via Averaging Theory." International Journal of Bifurcation and Chaos 30, no. 04 (2020): 2050051. http://dx.doi.org/10.1142/s0218127420500510.

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Consider the class of reversible quadratic systems [Formula: see text] with [Formula: see text]. These quadratic polynomial differential systems have a center at the point [Formula: see text] and the circle [Formula: see text] is one of the periodic orbits surrounding this center. These systems can be written into the form [Formula: see text] with [Formula: see text]. For all [Formula: see text] we prove that the averaging theory up to seventh order applied to this last system perturbed inside the whole class of quadratic polynomial differential systems can produce at most two limit cycles bif

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38

Lee, Joohee, та Yoon Kyung Park. "Evaluation of the convolution sums ∑a1m1+a2m2+a3m3+a4m4=nσ(m1)σ(m2)σ(m3)σ(m4) with lcm(a1,a2,a3,a4) ≤ 4". International Journal of Number Theory 13, № 08 (2017): 2155–73. http://dx.doi.org/10.1142/s1793042117501160.

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The generating functions of divisor functions are quasimodular forms of weight 2 and the product of them is a quasimodular form of higher weight. In this work, we evaluate the convolution sums [Formula: see text] for the positive integers [Formula: see text], and [Formula: see text] with lcm[Formula: see text]. We reprove the known formulas for the number of representations of a positive integer [Formula: see text] by each of the quadratic forms [Formula: see text] as an application of the new identities proved in this paper.

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39

Llibre, Jaume, and Regilene Oliveira. "Quadratic systems with an invariant conic having Darboux invariants." Communications in Contemporary Mathematics 20, no. 04 (2018): 1750033. http://dx.doi.org/10.1142/s021919971750033x.

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The complete characterization of the phase portraits of real planar quadratic vector fields is very far from being accomplished. As it is almost impossible to work directly with the whole class of quadratic vector fields because it depends on twelve parameters, we reduce the number of parameters to five by using the action of the group of real affine transformations and time rescaling on the class of real quadratic differential systems. Using this group action, we obtain normal forms for the class of quadratic systems that we want to study with at most five parameters. Then working with these

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40

Krumm, David. "A local–global principle in the dynamics of quadratic polynomials." International Journal of Number Theory 12, no. 08 (2016): 2265–97. http://dx.doi.org/10.1142/s1793042116501360.

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Let [Formula: see text] be a number field, [Formula: see text] a quadratic polynomial, and [Formula: see text]. We show that if [Formula: see text] has a point of period [Formula: see text] in every non-archimedean completion of [Formula: see text], then [Formula: see text] has a point of period [Formula: see text] in [Formula: see text]. For [Formula: see text] we show that there exist at most finitely many linear conjugacy classes of quadratic polynomials over [Formula: see text] for which this local–global principle fails. By considering a stronger form of this principle, we strengthen glob

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41

Yackel, Erna. "A Foundation for Algebraic Reasoning In the Early Grades." Teaching Children Mathematics 3, no. 6 (1997): 276–80. http://dx.doi.org/10.5951/tcm.3.6.0276.

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For many adults, algebra means solving systems of equations; finding the value of an unknown; using the quadratic formula; or otherwise working within a system of formulas, equations, and literal symbols.

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42

Earnest, A. G., and Ji Young Kim. "Integral quadratic forms avoiding arithmetic progressions." International Journal of Number Theory 16, no. 10 (2020): 2141–48. http://dx.doi.org/10.1142/s1793042120501109.

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For every positive integer [Formula: see text], it is shown that there exists a positive definite diagonal quaternary integral quadratic form that represents all positive integers except for precisely those which lie in [Formula: see text] arithmetic progressions. For [Formula: see text], all forms with this property are determined.

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43

Llibre, Jaume, Bruno D. Lopes, and Paulo R. da Silva. "Bifurcations of the Riccati Quadratic Polynomial Differential Systems." International Journal of Bifurcation and Chaos 31, no. 06 (2021): 2150094. http://dx.doi.org/10.1142/s0218127421500942.

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In this paper, we characterize the global phase portrait of the Riccati quadratic polynomial differential system [Formula: see text] with [Formula: see text], [Formula: see text] nonzero (otherwise the system is a Bernoulli differential system), [Formula: see text] (otherwise the system is a Liénard differential system), [Formula: see text] a polynomial of degree at most [Formula: see text], [Formula: see text] and [Formula: see text] polynomials of degree at most 2, and the maximum of the degrees of [Formula: see text] and [Formula: see text] is 2. We give the complete description of the phas

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Yan, Litan, Junfeng Liu, and Chao Chen. "The generalized quadratic covariation for fractional Brownian motion with Hurst index less than 1/2." Infinite Dimensional Analysis, Quantum Probability and Related Topics 17, no. 04 (2014): 1450030. http://dx.doi.org/10.1142/s0219025714500301.

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In this paper, we study the generalized quadratic covariation of f(BH) and BH defined by [Formula: see text] in probability, where f is a Borel function and BH is a fractional Brownian motion with Hurst index 0 < H < 1/2. We construct a Banach space [Formula: see text] of measurable functions such that the generalized quadratic covariation exists in L2(Ω) and the Bouleau–Yor identity takes the form [Formula: see text] provided [Formula: see text], where [Formula: see text] is the weighted local time of BH. These are also extended to the time-dependent case, and as an application we give

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45

Schwarzweller, Christoph, and Agnieszka Rowińska-Schwarzweller. "Quadratic Extensions." Formalized Mathematics 29, no. 4 (2021): 229–40. http://dx.doi.org/10.2478/forma-2021-0021.

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Summary In this article we further develop field theory [6], [7], [12] in Mizar [1], [2], [3]: we deal with quadratic polynomials and quadratic extensions [5], [4]. First we introduce quadratic polynomials, their discriminants and prove the midnight formula. Then we show that - in case the discriminant of p being non square - adjoining a root of p’s discriminant results in a splitting field of p. Finally we prove that these are the only field extensions of degree 2, e.g. that an extension E of F is quadratic if and only if there is a non square Element a ∈ F such that E and ( F a F\sqrt a ) ar

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46

Lord, Nick. "75.14 The Mechanics of the Quadratic Formula." Mathematical Gazette 75, no. 472 (1991): 197. http://dx.doi.org/10.2307/3620256.

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Masiero, Federica. "A Bismut–Elworthy formula for quadratic BSDEs." Stochastic Processes and their Applications 125, no. 5 (2015): 1945–79. http://dx.doi.org/10.1016/j.spa.2014.12.003.

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SHAVGULIDZE, KETEVAN. "ON THE NUMBER OF REPRESENTATIONS OF INTEGERS BY THE SUMS OF QUADRATIC FORMS $x_1^2 + x_1x_2 + 3x_2^2$." International Journal of Number Theory 05, no. 03 (2009): 515–25. http://dx.doi.org/10.1142/s1793042109002201.

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Izhakian, Zur, Manfred Knebusch, and Louis Rowen. "Supertropical quadratic forms II: Tropical trigonometry and applications." International Journal of Algebra and Computation 28, no. 08 (2018): 1633–76. http://dx.doi.org/10.1142/s021819671840012x.

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This paper is a sequel of [Z. Izhakian, M. Knebusch and L. Rowen, Supertropical semirings and supervaluations, J. Pure Appl. Algebra 220(1) (2016) 61–93], where we introduced quadratic forms on a module [Formula: see text] over a supertropical semiring [Formula: see text] and analyzed the set of bilinear companions of a quadratic form [Formula: see text] in case the module [Formula: see text] is free, with fairly complete results if [Formula: see text] is a supersemifield. Given such a companion [Formula: see text], we now classify the pairs of vectors in [Formula: see text] in terms of [Formu

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50

He, Zilong. "Inequalities for inert primes and their applications." International Journal of Number Theory 16, no. 08 (2020): 1819–32. http://dx.doi.org/10.1142/s1793042120500943.

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For any given non-square integer [Formula: see text], we prove Euclid’s type inequalities for the sequence [Formula: see text] of all primes satisfying the Kronecker symbol [Formula: see text], [Formula: see text] and give a new criterion on a ternary quadratic form to be irregular as an application, which simplifies Dickson and Jones’s argument in the classification of regular ternary quadratic forms to some extent.

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