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This theorem states that, if there are 2 equal chords in a circle, the angles formed by the equal chords at the center of the circle are equal. Given AB and CD are two equal chords in a circle with center O. To Prove ∠AOB = ∠COD Proof C
whereBis the unit ball inR4,wβ(x)=(loge|x|)βorwβ(x)=(log(1|x|)βforβ∈(0,1),2<p<4andf:R→Ris a superlinear continuous class function with supercritical exponential growth. We utilize the Nehari manifold method to establish the existence result. 1Introduction In this wo...
Further suppose that whenf(x)is divided by a linear polynomialp(x) = ( x -a), the quotient isq(x)and the remainder isr(x). In other words ,f(x)andp(x)are two polynomials such that the degree off(x) degree of p(x)andp(x) 0then we can find polynomialsq(x)andr(x)such tha...
The equations governing the motion of a three-dimensional liquid drop moving freely in an unbounded liquid reservoir under the influence of a gravitational
3, we prove that almost all triangular states can be uniquely determined by their RDMs on subsystems HA ⊗ HBHA ⊗ HB and HA ⊗ HCHA ⊗ HC. The final proof is in sect. sect. 4, which claims that the set of states that cannot be transformed into a triangular form, or that ...
In particular, we establish the Roundabout theorem which relates oscillatory properties of this equation to the solvability of the associated Riccati-type dynamic equation and to the positivity of the corresponding energy functional. This result is then used to prove (non)oscillation criteria for the ...
whereVis a non-symmetric bounded potential,ais an indefinite weight,0<q<1andf≠0is a nonnegative perturbation such thatf∈L2(RN)∩L2NN+2(RN). Using variational methods, we prove the existence of two solutions with negative and positive energies, one of these solutions being nonnegative. ...
(9) In this way, there exists a non-negative function R(t) such that C 0 Dαt V¯ (t , x ) + γ3 γ2 V¯ (t, x) + R (t ) = 0 (10) Taking into account the existence and uniqueness theorem for fractional-order differential equations (see [20]), it follows that V...
Theorem 8 The optimal control law for the linear system has the property of certainty equivalence if and only if it has no dual effect.The following lemma provides a sufficient condition for the no dual effect, which can be used to judge whether a control action with certainty equivalence is...