共查询到20条相似文献,搜索用时 15 毫秒
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H. K. Pathak A. S. Grewal 《International Journal of Mathematical Education in Science & Technology》2013,44(1):150-156
This note gives geometrical/graphical methods of finding solutions of the quadratic equation ax 2 + bx + c = 0, a p 0, with non-real roots. Three different cases which give rise to non-real roots of the quadratic equation have been discussed. In case I a geometrical construction and its proof for finding the solutions of the quadratic equation ax 2 + bx + c = 0, a p 0, when a,b,c ] R, the set of real numbers, are presented. Case II deals with the geometrical solutions of the quadratic equation ax 2 + bx + c = 0, a p 0, when b ] R, the set of real numbers; and a,c ] C, the set of complex numbers. Finally, the solutions of the quadratic equation ax 2 + bx + c = 0, a p 0, when a,c ] R, the set of real numbers, and b ] C, the set of complex numbers, are presented in case III. 相似文献
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H. K. Pathak A. S. Grewal 《International Journal of Mathematical Education in Science & Technology》2013,44(4):575-583
A general cubic equation ax 3 + bx 2 + cx + d = 0 where a, b, c, d ∈R, a ≠ 0 has three roots with two possibilities—either all three roots are real or one root is real and the remaining two roots are imaginary. Dealing with the second possibility this paper attempts to give the geometrical locations of the imaginary roots of the equation under three different sets of conditions. These sets of conditions include: (i) the real root of the given cubic equation is given, (ii) the real part of an imaginary root is given, and (iii) the imaginary part of an imaginary root is given. 相似文献
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F. M. Mukhamedov B. A. Omirov M. Kh. Saburov K. K. Masutova 《Siberian Mathematical Journal》2013,54(3):501-516
We give a criterion for the existence of solutions to an equation of the form x 3 + ax = b, where a, b ∈ ? p , in p-adic integers for p > 3. Moreover, in the case when the equation x 3 + ax = b is solvable, we give necessary and sufficient recurrent conditions on a p-adic number x ∈ ?* p under which x is a solution to the equation. 相似文献
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A. Schinzel 《Proceedings of the Steklov Institute of Mathematics》2012,276(1):250-256
A necessary and sufficient condition is given for an equation ax
m
+ bx
n
+ c = dy
p
+ ey
q
to have infinitely many rational solutions with a bounded denominator, under the assumption that m > n > 0, p > q > 0, ab ≠ 0 ≠ de and either m > p > 2, or m = p > 2 and n ≥ q. In a previous paper there was an additional assumption (m, n) = (p, q) = 1. 相似文献
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We show that there exist a C1function,f, of two variables and a setE ⊆ R2of zero Lebesgue measure such that using the natural three-dimensional parametrization of planesz = ax + by + ctangent to the surfacez = f(x, y), the (three-dimensional) interior of the set of parameter values, (a, b, c), of tangent planes corresponding to points (x, y) inEis nonempty. From the Morse–Sard theorem it follows that there are no such C2functions. We also study briefly the relationship of our example with the Denjoy–Young–Saks theorem. 相似文献
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G. B. Shabat 《Journal of Mathematical Sciences》2006,135(5):3420-3424
The paper is devoted to an elementary Diophantine problem motivated by Grothendieck’s dessins d’enfants theory. Namely, we consider the system of equations ax
j
+ by
j
+ cz
j
+ dt
j
= 0 (j = 1, 2, 3) with natural a, b, c, and d. For trivial reasons it has no real (hence rational) nonzero solutions; we study the cases where it has imaginary quadratic ones. We suggest an infinite family of such cases covering all the imaginary quadratic fields. We discuss this result from
the viewpoint of the Galois orbits of trees of diameter 4.
__________
Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 9, No. 3, pp. 229–236, 2003. 相似文献
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Yi Qian Wang 《数学学报(英文版)》2001,17(2):313-318
In this paper we consider the boundedness of all the solutions for the equation x″ + ax
+−bx
− = f(t) is a smooth 2π-periodic function, a and b are positive constants (a≠b).
Received November 27, 1998, Accepted October 15, 1999 相似文献
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Pingzhi Yuan 《Indagationes Mathematicae》2005,16(2):301-320
In this paper we investigate the solvability and the representation of the solutions of the equation ax2 +by2 = ckn. We extend and improve many known results. In particular, we completely solve the equation (a ± 1)x2 + (3a ? 1) = 4an, 2 ? n. 相似文献
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J. G. Byatt-Smith 《Studies in Applied Mathematics》1988,79(2):143-157
The properties of the solutions of the differential equation y″ = y2 ? x2 ? c subject to the condition that y is bounded for all finite x discussed. The arguments of Holmes and Spence have been used by Ockendon, Ockendon, and Johnson to show that there are no solutions if c is large and negative. Numerically we find that solutions exist provided c is greater than a critical value c* and estimate this value to be c* = ?…. As x tends to + ∞ the solutions are asymptotic to . The relation between A+ and ?+ are found analytically as A+ → ∞. This problem arises as a connection problem in the theory of resonant oscillations of water waves. 相似文献
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We provide additional methods for the evaluation of the integral
N0,4(a;m) : = ò0¥ \fracdx( x4 + 2ax2 + 1 )m+1,N_{0,4}(a;m) := \int_{0}^{\infty} \frac{dx}{( x^{4} + 2ax^{2} + 1 )^{m+1}}, 相似文献
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MA Asiru 《International Journal of Mathematical Education in Science & Technology》2013,44(5):695-699
Under predetermined conditions on the roots and coefficients, necessary and sufficient conditions relating the coefficients of a given cubic equation x 3?+?ax 2?+?bx?+?c?=?0 can be established so that the roots possess desired properties. In this note, the condition for one root of a cubic equation to be the negative reciprocal of another one is obtained. Given that the coefficients a, b, c of the cubic equation are in arithmetical or geometrical progression, further conditions are deived for one root to be the negative reciprocal of another. These results provide useful means for checking calculated roots of cubic equations and could serve the needs of teachers and students of Mathematical Sciences in tertiary institutions when the solution of cubic equations are first studied. 相似文献
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Let a, b, c be nonzero integers having no prime factors ≡ 3 (mod 4), not all of the same sign, abc squarefree, and for which Legendre's equation ax2 + by2 + cz2 = 0 is solvable in nonzero integers x, y, z. A property is proved yielding a congruence which must be satisfied by any solution x, y, z. 相似文献
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Let I be an interval of positive rational numbers. Then the set S (I) = T ∩ N, where T is the submonoid of (Q0+, +) generated by T, is a numerical semigroup. These numerical semigroups are called proportionally modular and can be characterized as the set of integer solutions of a Diophantine inequality of the form ax rood b 〈 cx. In this paper we are interested in the study of the maximal intervals I subject to the condition that S (I) has a given multiplicity. We also characterize the numerical semigroups associated with these maximal intervals. 相似文献
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Enrique González-Jiménez 《Acta Mathematica Hungarica》2014,142(1):231-243
We study solutions of the Markoff–Rosenberger equation ax 2+by 2+cz 2=dxyz whose coordinates belong to the ring of integers of a number field and form a geometric progression. 相似文献
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We completely describe the Siegel discs and attractors for the p-adic dynamical system f(x) = x
2n+1
+ axn+1
on the space of complex p-adic numbers. 相似文献
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Mao Hua LE 《数学学报(英文版)》2008,24(6):917-924
Let a, b and c be fixed coprime positive integers. In this paper we prove that if a^2 + b^2 = c^3 and b is an odd prime, then the equation a^x + b^y = c^z has only the positive integer solution (x, y, z) = (2,2,3). 相似文献
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