Electronic Journal of Differential Equations, Vol. 2025 (2025), No. 33, pp. 1–27. ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu DOI: 10.58997/ejde.2025.33 SPHERICAL COMPACTIFICATIONS OF CENTRAL FORCE EQUATIONS HARRY GINGOLD, JOCELYN QUAINTANCE Abstract. A spherical compactification is a map between an unbounded set of Rn and a bounded set on a sphere in Rn+1. This article rigorously defines a parameterized family of spherical compactifcations and applies such compactifications to systems and solutions of ordi- nary differential equations (ODEs) associated with central force equations. Spherical compact- ification provides a means of embedding Rn into a complete metric space. The compactified differential equation may have critical points that represent “critical points at infinity” of the original equation. These “critical points at infinity” in Rn may be appropriately labeled by ∞U , where U is a unit vector in Rn, and are “visualized” as points on the rim of a spherical compactifaction. To further legitimize objects of the form ∞U , we develop a new calculus which interprets objects of the form ∞U1 + ∞U2. We then utilize these spherical compactifications, which are of the form w(t) = θ−1(t)z(t), to transform a first order vector valued differential equation w′(t) = F (w(t)) into the first order vector valued differential equation z′(t) = H(z(t)) and provide two theorems which manifest the correspondence between finite critical points of w′(t) = F (w(t)) and z′(t) = H(z(t)). 1. Introduction The overarching purpose of this article is to rigorously fill in gaps in the compactification methods for ordinary differential equations (ODEs). The secondary purpose is to apply such compactification methods to central force equations, especially Kepler’s problem. Recall that central force equations model the motion of a particle under a central force field F (t) ∈ R3. If r(t) ∈ R3 is the position of the particle, the central force equations are given by F (t) = F (r) r(t) ∥r(t)∥ , F (r) ∈ R. (1.1) The Newtonian laws of motion F (t) = d dt (mr′(t)) = ma(t), (1.2) are special cases of (1.1) since the laws of gravitation imply that a(t) := r′′(t) = K ∥r∥3 r(t), K ∈ R. (1.3) Observe that (1.1) and (1.2) insinuate that F (r) = mK/∥r(t)∥2. A compactification is a continuous mapping that maps an unbounded set of Rn into a bounded set of Rd. The concept of using compactification to visualize “objects at infinity” was known to the ancient Greek astronomer-mathematician Ptolemy (circa 100 to 170, C.E.) in the form of stereographic projection [54, Section 3.6]. Compactifications methods, when applied to ODEs, take an unbounded set of solutions into a bounded set of solutions. Many books prefer to apply Poincaré compactifaction to solutions of ODEs [54, Section 3.10],. The Poincaré compactification 2020 Mathematics Subject Classification. 70F15, 85A04. Key words and phrases. Central force equations; Newton’s celestial mechanics equations; N -body problem; finite critical point; critical point at infinity; spherical compactifications; Kepler’s problem; stereographic projection; ultra-extended Rn; metric analysis. ©2025. This work is licensed under a CC BY 4.0 license. Submitted November 25, 2024. Published April 3, 2025. 1 2 H. GINGOLD, J. QUAINTANCE EJDE-2025/33 maps the hyperplane of Rn+1 with the equation xn+1 = 0 (itself homeomorphic to Rn) onto the “hemisphere” Sn+1, where Sn+1 := {(x1, x2, . . . , xn, xn+1) T : n∑ i=1 x2 i + (xn+1 − 1)2 = 1, 0 ≤ xn+1 ≤ 1}. However, when using the Poincaré compactification, these books avoid “rim” of hemisphere and work with S̊n+1 := {(x1, x2, . . . , xn, xn+1) T : n∑ i=1 x2 i + (xn+1 − 1)2 = 1 0 ≤ xn+1 < 1}; see [13, 14, 54]. This is because they did not turn the hyperplane with equation xn+1 = 0 (or equivalently Rn) into a complete metric space. Theorem 3.8 herein shows how to turn Rn into a complete metric spaces and addresses this gap in the literature. Theorem 3.8 is an extension of the initial work done by Y. Gingold and H. Gingold in which they embed R2 into a complete metric space [25]. The compactified differential systems and equations may have critical points that represent “critical points at infinity” of the original (uncompactificed) systems and equations. These “critical points at infinity” in Rn may be appropriately labeled by ∞U , where U is a unit vector in Rn. Kepler’s problem, and more generally, Newton’s equations of celestial mechanics, are shown to have such “critical points at infinity” [24]. The existence of such “critical points at infinity” necessitates developing a calculus which interprets objects of the form ∞U1 +∞U2; see Theorem 2.9. Such a calculus, at least as far as we know, is not found in the literature. Section 2 provides the algebraic underpinning of this calculus and is the first step in legitimizing these objects. Further legitimization of ∞U requires the extension of Rn to a complete metric space UERn equipped with a proper metric given by Theorem 3.8. In this article we use a parametrized family of spherical compactifications to “compactify” these differential equations. This parametrized family [25] generalizes the stereographic and Poincaré projections by generating a spectrum of spherical compactifications of which the stereographic and Poincaré projections are special cases. We use a parametrized family of compactifications rather than just one spherical compactification for several reasons. When transforming w(t) = F (w(t)) into z(t) = H(z(t)), where w(t) = θ−1(t)z(t), there are critical points at infinity for z(t) = H(z(t)) which vary with the compactifications that are employed since these critical points are contained within {z(t) ∈ Rn : ∥z(t)∥ = √ 1− γ2}. Alternatively, a critical point obtained via compactification is not an invariant of the original equation w′ = F (w). What is invariant under the inverse transformation is the “direction of infinity”, namely that lim t→∞ w(t) ∥w(t)∥ = lim t→∞ z(t) ∥z(t)∥ = U. Secondly, the parametrized family brings out the fact that the stereographic projection is unfit for celestial mechanics as it leads to an unbounded compactified equation with the condition 1− γ2 = 0. We end this paper by utilizing spherical compactifications to transform a first order vector valued differential equation w′(t) = F (w(t)) into the first order vector valued differential equation z′(t) = H(z(t)). We then provide two theorems which manifest the correspondence between finite critical points of w′(t) = F (w(t)) and z′(t) = H(z(t)); see Theorem 4.3 and Theorem 4.11. 2. Algebraic structure of the ultra extended Rn In this section we provide the mathematical framework for the Ultra Extended Rn, herein referred to as UERn, whenever n is any fixed positive integer. We begin with the algebraic definition of ∞U . Definition 2.1. Let n be a fixed positive integer. For any constant U ∈ Rn such that U := (u1, u2, . . . , un) T , ∥U∥ = √ u2 1 + u2 2 + · · ·+ u2 n = UTU = 1, EJDE-2025/33 SPHERICAL COMPACTIFICATIONS 3 and for any (Vm)m∈N ⊂ Rn such that (a) lim m→∞ ∥Vm∥ = ∞, (b) lim m→∞ Vm ∥Vm∥ = U, we say Vm → ∞U (as m → ∞) and that (Vm)m∈N is an approximation sequence of ∞U . See Figure 1. U V V 1 2 V 3 O V n V n+1 UN Figure 1. Manifestation of ∞U via the approximation sequence (Vm)m∈N. Definition 2.2. Let Sn−1 denote the unit sphere in Rn centered at the origin, i.e. Sn−1 := {U ∈ Rn : ∥U∥ = 1}. The ideal set IDn associated with Rn is defined as IDn := {∞U : U ∈ Sn−1}, (2.1) while the ultra extended Rn, which we denote as UERn, is defined as UERn := IDn∪̇Rn. (2.2) We define the algebraic operations of addition and scalar multiplication on UERn. The defini- tion of these operations are provided by the following series of propositions. Proposition 2.3. Let U ∈ Sn−1. If (Vm)m∈N ⊂ Rn such that Vm → ∞U , and if k ∈ R/{0}, then kVm → ∞kU/|k|. Proof. Since Definition 2.1 implies that limm→∞ ∥Vm∥ = ∞ and that limm→∞ Vm ∥Vm∥ = U , we deduce that lim m→∞ kVm |k|∥Vm∥ = kU |k| , and the result follows. □ Proposition 2.3 justifies the following definition of scalar multiplication in UERn. 4 H. GINGOLD, J. QUAINTANCE EJDE-2025/33 Definition 2.4. Let X ∈ UERn and k ∈ R. If X ∈ Rn, then kX ∈ Rn. If X ∈ IDn, i.e. X = ∞U for U ∈ Sn−1, then k∞U := ∞kU |k| , k ∈ R/{0}. (2.3) In particular, −∞U = ∞(−U). Note that 0∞U is undefined. Next we define various additions operations in UERn. First a proposition which will be used to define V +∞U . Proposition 2.5. Let V ∈ Rn and U ∈ Sn−1. Let (Vm)m∈N ⊂ Rn such that Vm → ∞U . Then V + Vm → ∞U . Proof. By Definition 2.1 we know that limm→∞ Vm ∥Vm∥ = U with limm→∞ ∥Vm∥ = ∞. Since ∥V ∥ < ∞, and since ∥Vm∥ ≤ ∥V ∥ + ∥V + Vm∥, we deduce that limm→∞ ∥V + Vm∥ = ∞. Furthermore, since ∥Vm∥ ∥Vm∥+ ∥V ∥ ≤ ∥Vm∥ ∥V + Vm∥ ≤ ∥Vm∥ | ∥Vm∥ − ∥V ∥ | , the squeeze theorem implies that limm→∞ ∥Vm∥ ∥V+Vm∥ = 1. Then lim m→∞ V + Vm ∥V + Vm∥ = lim m→∞ V ∥V + Vm∥ + lim m→∞ Vm ∥V + Vm∥ = 0 + lim m→∞ Vm ∥Vm∥ lim m→∞ ∥Vm∥ ∥V + Vm∥ = U. □ Remark 2.6. In Proposition 2.5 the constant vector V ∈ Rn can be replaced with a vector function V (t) ∈ Rn such that for all t ∈ R, ∥V (t)∥ < M , where M is a fixed nonnegative real number independent of t. It remains to determine the meaning of ∞U + ∞Û . The proof of Proposition 2.3 shows for all fixed k > 0, (kVm)m∈N is an approximation sequence of ∞U . We extend this result by (αm)m∈N ⊂ R+ which satisfies lim infm→∞ αm > 0, where R+ := {x ∈ R : 0 < x < ∞}. Then given Vm → ∞U , since lim inf m→∞ αm lim m→∞ ∥Vm∥ ≤ lim m→∞ ∥αmVm∥, and since lim infm→∞ αm ∈ R+ implies that lim inf m→∞ αm lim m→∞ ∥Vm∥ = ∞, we find that lim m→∞ ∥αmVm∥ = ∞, lim m→∞ αmVm ∥αmVm∥ = lim m→∞ Vm ∥Vm∥ = U. (2.4) The calculations of (2.4) show that (αmVm)m∈N is also an approximation sequence of ∞U . Hence we may consider ∞U as an infinite family of approximation sequences. Definition 2.7. Let n be a fixed positive integer. For any U ∈ Sn−1, define ∞U via ∞U := { (Vm)m∈N ⊂ Rn : lim m→∞ ∥Vm∥ = ∞, lim m→∞ Vm ∥Vm∥ = U } := {(Vm)m∈N}U . (2.5) Equation (2.5) provides an alternative definition for ∞U . Every statement Vm → ∞U is equivalent to ∞U = {(Vm)m∈N}U , i.e. Vm → ∞U for all (Vm) ∈ {(Vm)m∈N}U . Hence we consider (Vm)m∈N as a representative of the set {(Vm)m∈N}U . Definition 2.8. Let p ∈ R+ and let U, Û ∈ Sn−1. We define Sp as Sp := {( (Vm), (Wm) ) : (Vm) ∈ {(Vm)m∈N}U , (Wm) ∈ {(Wm)m∈N}Û , lim m→∞ ∥Vm∥ ∥Wm∥ = p } , (2.6) where (Vm) := (Vm)m∈N ∈ {(Vm)m∈N}U and (Wm) := (Wm)m∈N ∈ {(Wm)m∈N}Û are defined via (2.5). We define S0 as S0 := { ((Vm), (Wm) ) : (Vm) ∈ {(Vm)m∈N}U , (Wm) ∈ {(Wm)m∈N}Û , lim m→∞ ∥Vm∥ ∥Wm∥ = 0 } , (2.7) EJDE-2025/33 SPHERICAL COMPACTIFICATIONS 5 and we define S∞ as S∞ := {( (Vm), (Wm) ) : (Vm) ∈ {(Vm)m∈N}U , (Wm) ∈ {(Wm)m∈N}Û , lim m→∞ ∥Vm∥ ∥Wm∥ = ∞ } . (2.8) The following theorem will be used to justify ∞U +∞Û = ∞VU,Û . Theorem 2.9. Let U, Û ∈ Sn−1. Assume that θ1U + (1− θ1)Û ̸= 0⃗ whenever 0 ≤ θ1 ≤ 1. (a) Let p ∈ R+ and let ((Vm), (Wm)) ∈ Sp. For every 0 < θ1 < 1, there exists a unique 0 < θ̂1 < 1 such that θ1Vm + (1− θ1)Wm → ∞ θ̂1U + (1− θ̂1)Û ∥θ̂1U + (1− θ̂1)Û∥ . (2.9) (b) For ((Vm), (Wm)) ∈ S0, let θ1 = 0 = θ̂1. Then θ1Vm + (1− θ1)Wm → ∞Û . (2.10) (c) For ((Vm), (Wm)) ∈ S∞, let θ1 = 1 = θ̂1. Then θ1Vm + (1− θ1)Wm → ∞U. (2.11) Proof. For (a) we need to prove that lim m→∞ ∥θ1Vm + (1− θ1)Wm∥ = ∞ lim m→∞ θ1Vm + (1− θ1)Wm ∥θ1Vm + (1− θ1)Wm∥ = θ̂1U + (1− θ̂1)Û ∥θ̂1U + (1− θ̂1)Û∥ . (2.12) To prove the first relation of (2.12), we put Ψm := θ1Vm + (1− θ1)Wm = θ1∥Vm∥ Vm ∥Vm∥ + (1− θ1)∥Wm∥ Wm ∥Wm∥ , (2.13) and observe that Ψm = ∥Wm∥ [ θ1 ∥Vm∥ ∥Wm∥ Vm ∥Vm∥ + (1− θ1) Wm ∥Wm∥ ] , Ψm = ∥Vm∥ [ θ1 Vm ∥Vm∥ + (1− θ1) ∥Wm∥ ∥Vm∥ Wm ∥Wm∥ ] . (2.14) Notice that both representations are well defined because (Vm)m∈N is an approximation se- quence for ∞U and (Wm)m∈N is an approximation sequence for ∞Û . Our goal is to show that limm→∞ ∥Ψm∥ = ∞. If we can show that there exists an m1 > 0 and a constant δ(m1) > 0 such that for all m ≥ m1 we have∥∥∥θ1 ∥Vm∥ ∥Wm∥ Vm ∥Vm∥ + (1− θ1) Wm ∥Wm∥ ∥∥∥ = δ(m1) > 0, (2.15) then since limm→∞ ∥Wm∥ = ∞, the first representation of Ψm in (2.14) will indeed imply that limm→∞ ∥Ψm∥ = ∞. We prove (2.15) via contradiction. Assume by contradiction that there exists a subsequence (Vmk) of (Vm) and a subsequence (Wmk) of (Wm) such that lim k→∞ ∥Vmk∥ ∥Wmk∥ = p, lim k→∞ [ θ1 ∥Vmk∥ ∥Wmk∥ Vmk ∥Vmk∥ + (1− θ1) Wmk ∥Wmk∥ ] = θ1pU + (1− θ1)Û = 0⃗. (2.16) By construction, 0 < p < ∞, and θ1pU + (1− θ1)Û = 0⃗ ⇐⇒ θ1pU + (1− θ1)Û θ1p+ 1− θ1 = 0⃗. (2.17) 6 H. GINGOLD, J. QUAINTANCE EJDE-2025/33 This statement is equivalent to θ1pU + (1− θ1)Û θ1p+ 1− θ1 = 0⃗ = θ̂1U + (1− θ̂1)Û , (2.18) where 0 < θ̂1 = θ1p θ1p+ 1− θ1 < 1, (2.19) which is a contradiction to the assumption that θ1U + (1− θ1)Û ̸= 0⃗ whenever 0 ≤ θ1 ≤ 1. Hence (2.15) is true, which in turn implies that the first relation of (2.12) holds. A similar proof shows that there exists an m2 > 0 and a constant δ(m2) > 0 such that for all m ≥ m2 we have ∥∥∥θ1 Vm ∥Vm∥ + (1− θ1) ∥Wm∥ ∥Vm∥ Wm ∥Wm∥ ∥∥∥ = δ(m2) > 0. (2.20) Now we focus on proving the second relation of (2.12). Equations (2.15) and (2.20), when combined with (2.14), show that the denominators in the following calculation are nonzero for large enough m. Recall that limm→∞ ∥Vm∥/∥Wm∥ = p ∈ R+. lim m→∞ θ1Vm + (1− θ1)Wm ∥θ1Vm + (1− θ1)Wm∥ = lim m→∞ θ1Vm ∥θ1Vm∥ lim m→∞ ∥θ1Vm∥ ∥θ1Vm + (1− θ1)Wm∥ + lim m→∞ (1− θ1)Wm ∥(1− θ1)Wm∥ lim m→∞ ∥(1− θ1)Wm∥ ∥(1− θ1)Wm + θ1Vm∥ = U lim m→∞ 1 ∥θ1Vm+(1−θ1)Wm∥ ∥θ1Vm∥ + Û lim m→∞ 1 ∥θ1Vm+(1−θ1)Wm∥ ∥(1−θ1)Wm∥ = U lim m→∞ 1∥∥ θ1Vm ∥θ1Vm∥ + (1−θ1)Wm ∥(1−θ1)Wm∥ ∥(1−θ1)Wm∥ ∥θ1Vm∥ ∥∥ + Û lim m→∞ 1∥∥ θ1Vm ∥θ1Vm∥ ∥θ1Vm∥ ∥(1−θ1)Wm∥ + (1−θ1)Wm ∥(1−θ1)Wm∥ ∥∥ = U ∥U + 1−θ1 pθ1 Û∥ + Û ∥ pθ1 1−θ1 U + Û∥ . (2.21) The calculation of (2.21) implies that lim m→∞ θ1Vm + (1− θ1)Wm ∥θ1Vm + (1− θ1)Wm∥ = pθ1U + (1− θ1)Û ∥pθ1U + (1− θ1)Û∥ = pθ1 pθ1+(1−θ1) U + 1−θ1 pθ1+(1−θ1) Û ∥ pθ1 pθ1+(1−θ1) U + 1−θ1 pθ1+(1−θ1) Û∥ , and we complete the proof by setting θ̂1 := pθ1 pθ1 + 1− θ1 = pθ1 (p− 1)θ1 + 1 . (2.22) Proof of (b). Since ((Vm), (Wm)) ∈ S0, a carefully reading of the proof of (2.15) shows that it is still valid for p = 0 as long as θ1 ̸= 1. Thus we can choose θ1 = 0 = θ̂1 and obtain (2.10). The choice of θ1 = 0 is justified by assuming that p = 0 in (2.21) and θ1 ̸= 1, in which case we obtain lim m→∞ θ1Vm + (1− θ1)Wm ∥θ1Vm + (1− θ1)Wm∥ = Û , (2.23) since ∥∥U + 1− θ1 θ1p Û ∥∥ = ∞. (2.24) EJDE-2025/33 SPHERICAL COMPACTIFICATIONS 7 For the proof of (c), we make the following adjustments to the proof of (2.15). We choose a subsequence (Vmk) of (Vm) and a subsequence (Wmk) of (Wm) such that lim k→∞ ∥Vmk∥ ∥Wmk∥ = ∞, lim k→∞ ∥∥∥θ1 ∥Vmk∥ ∥Wmk∥ Vmk ∥Vmk∥ + (1− θ1) Wmk ∥Wmk∥ ∥∥∥ = 0. (2.25) We can see the contradiction by noticing that∥∥∥θ1 ∥Vmk∥ ∥Wmk∥ Vmk ∥Vmk∥ + (1− θ1) Wmk ∥Wmk∥ ∥∥∥ ≥ ∥θ1 ∥Vmk∥ ∥Wmk∥ Vmk ∥Vmk∥ ∥ − ∥(1− θ1) Wmk ∥Wmk∥ ∥ ≥ θ1 ∥Vmk∥ ∥Wmk∥ − (1− θ1). (2.26) By taking the limit of (2.26), as long as θ1 ̸= 0, we find that 0 = lim k→∞ ∥θ1 ∥Vmk∥ ∥Wmk∥ Vmk ∥Vmk∥ + (1− θ1) Wmk ∥Wmk∥ ∥ ≥ θ1 lim k→∞ ∥Vmk∥ ∥Wmk∥ − (1− θ1) = ∞, (2.27) which is an obvious contradiction. Thus we can choose θ1 = 1 = θ̂1 and obtain (2.11). The choice of θ1 = 1 is justified by assuming that p = ∞ and θ1 ̸= 0 in (2.21), in which case we obtain lim m→∞ θ1Vm + (1− θ1)Wm ∥θ1Vm + (1− θ1)Wm∥ = U, (2.28) since ∥∥ pθ1 1− θ1 U + Û ∥∥ = ∞. (2.29) □ Remark 2.10. Equation (2.9) shows that not only are all 0 ≤ θ1 ≤ 1 attained but also that additional values of 0 ≤ θ1 ≤ 1 are not possible. Also observe that given U, Û ∈ Sn−1, if there exists a nonzero vector N ∈ Rn such that ⟨N,U⟩ and ⟨N, Û⟩ are of the same sign, then ⟨N, θ1U + (1− θ1)Û⟩ ≠ 0, 0 ≤ θ1 ≤ 1, and consequently θ1U + (1− θ1)Û ̸= 0̂ as desired. When deriving (2.21), since 0 < θ1 < 1 and 0 < p < ∞, we implicitly use that ∥U + 1− θ1 θ1p Û∥ ≠ 0 ⇐⇒ ∥θ1pU + (1− θ1)Û∥ ≠ 0 ⇐⇒ ∥∥ θ1pU pθ1 + 1− θ1 + (1− θ1)Û pθ1 + 1− θ1 ∥∥ ̸= 0, (2.30) and that∥∥ θ1p 1− θ1 U + Û ∥∥ ̸= 0 ⇐⇒ ∥θ1pU +(1− θ1)Û∥ ≠ 0 ⇐⇒ ∥∥ θ1pU pθ1 + 1− θ1 + (1− θ1)Û pθ1 + 1− θ1 ∥∥ ̸= 0. (2.31) Since ∥θ1pU + (1− θ1)Û∥ ≠ 0 ⇐⇒ ∥θ1pU + (1− θ1)Û∥2 ̸= 0, and ∥θ1pU + (1− θ1)Û∥2 = ⟨θ1pU + (1− θ1)Û , θ1pU + (1− θ1)Û⟩ = (θ1p) 2∥U∥2 + (1− θ1) 2∥Û∥2 + 2θ1(1− θ1)p⟨U, Û⟩ = (θ1p) 2 + (1− θ1) 2 + 2θ1(1− θ1)p⟨U, Û⟩, (2.32) if ⟨U, Û⟩ > 0, then ∥θ1pU + (1− θ1)Û∥2 ̸= 0. We use ⟨U, Û⟩ > 0 to replace θ1U + (1− θ1)Û ̸= 0⃗ whenever 0 ≤ θ1 ≤ 1 as follows. Proposition 2.11. Let U, Û ∈ Sn−1, where ⟨U, Û⟩ > 0. 8 H. GINGOLD, J. QUAINTANCE EJDE-2025/33 (a) Let p ∈ R+ and let ((Vm), (Wm)) ∈ Sp. For every 0 < θ1 < 1, there exists a unique 0 < θ̂1 < 1 such that θ1Vm + (1− θ1)Wm → ∞ θ̂1U + (1− θ̂1)Û ∥θ̂1U + (1− θ̂1)Û∥ . (2.33) (b) For ((Vm), (Wm)) ∈ S0, let θ1 = 0 = θ̂1. Then θ1Vm + (1− θ1)Wm → ∞Û . (2.34) (c) For ((Vm), (Wm)) ∈ S∞, let θ1 = 1 = θ̂1. Then θ1Vm + (1− θ1)Wm → ∞U. (2.35) Proof. By assumption Vm → ∞U and Wm → ∞Û . This implies that lim m→∞ 〈 Vm ∥Vm∥ , Wm ∥Wm∥ 〉 = ⟨U, Û⟩, and hence there exists m1 > 0 and ϵ(m1) > 0 such that〈 Vm ∥Vm∥ , Wm ∥Wm∥ 〉 > ϵ(m1) > 0, whenever m ≥ m1. (2.36) Since limm→∞ ∥Vm∥ = ∞ and limm→∞ ∥Wm∥ = ∞, inequality (2.36) implies that ⟨Vm,Wm⟩ > ∥Vm∥∥Wm∥ϵ(m1) > 0, whenever m ≥ m1. (2.37) We are now in a position to prove (a). Once again we need to verify the two limits of (2.12). Observe that for n ≥ m1, Inequality (2.37) implies that ∥θ1Vm + (1− θ1)Wm∥2 = ⟨θ1Vm + (1− θ1)Wm, θ1Vm + (1− θ1)Wm⟩ = θ21∥Vm∥2 + (1− θ1) 2∥Wm∥2 + 2θ1(1− θ1)⟨Vm,Wm⟩ > θ21∥Vm∥2 + (1− θ1) 2∥Wm∥2. (2.38) Inequality (2.38) implies that lim m→∞ ∥θ1Vm + (1− θ1)Wm∥2 > θ21 lim m→∞ ∥Vm∥2 + (1− θ1) 2 lim m→∞ ∥Wm∥2 = ∞, which shows that limm→∞ ∥θ1Vm + (1 − θ1)Wm∥ = ∞, which is the first limit of (2.12). The calculations of (2.21) and (2.22) are still valid and prove the second limit of (2.12). The proof of Part (b) follows from (2.23) and (2.24), while the proof of Part (c) follows from (2.28) and (2.29). □ Theorem 2.9 implies that we set ∞U +∞Û = { ∞V : V = θ1U + (1− θ1)Û ∥θ1U + (1− θ1)Û∥ , 0 ≤ θ1 ≤ 1 } (2.39) whenever θ1U + (1− θ1)Û ̸= 0⃗, for 0 ≤ θ ≤ 1. We propose to use the notation ∞VU,Û := { ∞V : V = θ1U + (1− θ1)Û ∥θ1U + (1− θ1)Û∥ , 0 ≤ θ1 ≤ 1 } (2.40) to represent the right side of (2.39). Proposition 2.5 and Theorem 2.9 justify the following defini- tion of additions in UERn. Definition 2.12. Let V,W ∈ Rn, and let U, Û ∈ Sn−1 such that θ1U + (1− θ1)Û ̸= 0⃗, whenever 0 ≤ θ1 ≤ 1. Then V +W ∈ Rn and (a) V +∞U := ∞U , (b) ∞U +∞Û := ∞VU,Û . Remark 2.13. Formula (2.40) immediately shows that ∞U +∞Û = ∞Û +∞U . EJDE-2025/33 SPHERICAL COMPACTIFICATIONS 9 3. Spherical compactification of UERn We now extend the construction in [25, 27] and derive a compactification of UERn as a spherical bowl of Sn, where Sn = {U ∈ Rn+1 : ∥U∥ = 1}, and ∥ · ∥ is the Euclidean norm in Rn+1. This construction provides a geometric realization of IDn as points in Sn and allows us to turn UERn into a complete metric space. Definition 3.1. Let γ be a fixed positive number with 0 < γ < 1. The spherical bowl associated with γ, namely SBn γ , is defined as SBn γ := {(x1, x2, . . . , xn+1) T ∈ Sn : −1 ≤ xn+1 ≤ γ}. (3.1) The open spherical bowl is defined as OSBn γ := {(x1, x2, . . . , xn+1) T ∈ Sn : −1 ≤ xn+1 < γ}. (3.2) The “open upper hemisphere” of SBn γ is defined as SBn,+ γ := {Z = (x1, x2, . . . , xn+1) T ∈ SBn γ : 0 < xn+1 < γ}. (3.3) The “open lower hemisphere” of SBn γ is defined as SBn,− γ := {Z = (x1, x2, . . . , xn+1) T ∈ SBn γ : xn+1 < 0}. (3.4) The boundary or “rim” of the spherical bowl is defined as SBn γ /OSBn γ := {(x1, x2, . . . , xn+1) T ∈ Sn : xn+1 = γ}. (3.5) See Figure 2. P = (0,0,γ ) (0,0,-1) (0,0,1) (1,0,0) (0,1,0) (1,0,0) ) (1,0,0) ) ) ) ) ) )P =P =P =P = (0,0, (0,0,P = (0,0, (0,0,P = (0,0, (0,0,γγγγ ) ) ) )γγγγγ ) ) ) ) ) ) ) ) )γγ )γγγ ) ) ) ) )γγ ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) (1,0,0)(1,0,0)(1,0,0) Figure 2. Spherical bowl in R3. Remark 3.2. We illustrate the compactification of UER2 onto a spherical bowl SB2 γ = {X ∈ R3 : ∥X∥ = 1, −1 ≤ z ≤ γ}, where the vectors are written in row form with the transpose notation suppressed. 10 H. GINGOLD, J. QUAINTANCE EJDE-2025/33 Remark 3.3. Definition 3.1 can be extended to the case of γ = 1. In this case the compactifi- cation coincides with the stereographic projection; see [16] and [25]. We will not work with the stereographic projection since it does not preserve directions at infinity. Also the stereographic projection, unlike the spherical projections for 0 < γ < 1 of Definition 3.6, fails to locate critical points for the compactified equations of the expanding universe. We now describe a radial projection between OSBn γ and a copy of Rn embedded in Rn+1, which we denote as R̃n; see (3.6). To construct the radial lines of this projection, since OSBn γ ⊂ Rn+1, we need to use R̃n. However, since Rn is homeomorphic to R̃n, we use Rn as the domain of the bijective map; see Proposition 3.5. We define R̃n := {(x1, x2, . . . , xn, xn+1) T ∈ Rn+1 : xn+1 = 0}. (3.6) Let Z = (x1, x2, . . . , xn, xn+1) T ∈ OSBn γ , and let Q = (q1, q2, . . . , qn, 0) T ∈ R̃n. We define the projection point P as P := (0, 0, . . . , γ)T ∈ Rn+1. (3.7) Since xn+1 is determined from (xi) n i=1, i.e. x 2 n+1 = 1− ∑n i=1 x n i , and since Proposition 3.5 will only depend of the first n coordinates of Q, we will emphasize the importance of the first n coordinates in Z and Q by defining Z = (Z̃, xn+1) T ∈ OSBn γ , Z̃ := (x1, x2, . . . , xn) T Q = (Q, 0)T ∈ R̃n, Q := (q1, q2, . . . , qn) T ∈ Rn. (3.8) Next define the nonnegative real valued quantities R2 := x2 1 + x2 2 + · · ·+ x2 n = Z̃T Z̃, r2 := QTQ = QTQ = q21 + q22 + · · ·+ q2n. (3.9) We require that P , Z, and Q be collinear in a positive direction, namely that P⃗Z = θP⃗Q ⇐⇒ θ−1P⃗Z = P⃗Q, 0 < θ. (3.10) See Figure 3. (0,0,-1) (0,0,1) (1,0,0) (0,1,0) (1,0,0)(1,0,0)(1,0,0) 2(x , x , x )1 3Z = (q , q , 0)Q = radius = (1 - γ )γ2 ) ) 1/2 radius = (1 - 1 2 P = (0,0,γ ) Figure 3. Relationship between P , Z, and Q in R3. Equation (3.10) implies that xi = θqi ⇐⇒ qi = xi θ , 1 ≤ i ≤ n, xn+1 = (1− θ)γ. (3.11) EJDE-2025/33 SPHERICAL COMPACTIFICATIONS 11 The equation for xn+1 of (3.11) shows that if 0 < θ < 1, then Z ∈ SBn,+ γ , while if θ > 1, then Z ∈ SBn,− γ . The left relations of (3.11) imply that if 0 < θ < 1, then Q ∈ B+ n,n+1 := {(q1, q2, . . . , qn, 0)T ∈ R̃n : q21 + q22 + · · · q2n > 1}, while if θ > 1, then Q ∈ B− n,n+1 := {(q1, q2, . . . , qn, 0)T ∈ R̃n : q21 + q22 + · · · q2n < 1}. Thus, the radial projection of (3.10) maps SBn,+ γ to B+ n,n+1, and it maps SBn,− γ to B− n,n+1; see Figure 3. To show that radial projection of (3.10) is a continuous bijection, given Q ∈ R̃n, we uniquely solve for θ in order to determine the corresponding Z ∈ OSBn γ . Recall that Z ∈ SBn γ , which means x2 1 + x2 2 + · · ·+ x2 n+1 = 1. (3.12) Substitute (3.11) in (3.12) to obtain θ2q21 + θ2q22 + · · ·+ θ2q2n + (1− θ)2γ2 = 1, which is equivalent to θ2(q21 + q22 + · · ·+ q2n + γ2)− 2θγ2 + γ2 − 1 = θ2(r2 + γ2)− 2θγ2 + γ2 − 1 = 0. (3.13) Thus θ = 2γ2 ± √ 4γ4 − 4(r2 + γ2)(γ2 − 1) 2(r2 + γ2) = γ2 ± √ γ2 + (1− γ2)r2 r2 + γ2 , and since we require that θ > 0, we choose θ = γ2 + √ γ2 + (1− γ2)r2 r2 + γ2 . (3.14) Conversely given Z ∈ OSBn γ , we uniquely solve for θ in order to determine the corresponding Q ∈ R̃n. We use (3.12) and substitute in the relationship for xn+1 given by (3.11) to obtain x2 1 + x2 2 + · · ·+ x2 n + (1− θ)2γ2 = R2 + (1− θ)2γ2 = 1, which implies that θ = 1± √ 1−R2 γ . (3.15) If Z ∈ SBn,+ γ , i.e. xn+1 > 0, then 0 < θ < 1, and we choose θ = 1− √ 1−R2 γ . (3.16) If Z ∈ SBn,− γ , i.e. xn+1 < 0, then θ > 1, and we choose θ = 1 + √ 1−R2 γ . (3.17) If θ = 1, then R2 = 1, and Z = Q. Remark 3.4. Observe that (3.16) and (3.17) are undefined when γ = 0. This degeneracy is why we required 0 < γ < 1. Since Rn is homeomorphic to R̃n with the induced topology, the above calculations prove the following proposition. Proposition 3.5. Given P = (0, 0, . . . , γ)T ∈ Rn+1, where 0 < γ < 1, for any Q = (q1, q2, . . . , qn) T ∈ Rn define G : Rn → OSBn γ as G(Q) = (θq1, θq2, . . . , θqn, (1− θ)γ)T , θ = γ2 + √ γ2 + (1− γ2)QTQ QTQ+ γ2 . (3.18) 12 H. GINGOLD, J. QUAINTANCE EJDE-2025/33 For each Z = (x1, x2, . . . , xn, xn+1) T ∈ OSBn γ , we define L̂ : OSBn γ → Rn via L̂(Z) =  γ γ− √ 1− ∑n i=1 x2 i (x1, x2, . . . , xn) T , Z ∈ SBn,+ γ γ γ+ √ 1− ∑n i=1 x2 i (x1, x2, . . . , xn) T , Z ∈ SBn,− γ (x1, x2, . . . , xn) T , ∑n i=1 x 2 i = 1 ⇐⇒ xn+1 = 0. (3.19) If OSBn γ inherits the induced topology from Rn+1, then G is a continuous bijection with G−1 ≡ L̂. To obtain a bijection between IDn and SBn γ /OSBn γ , we need a limiting argument to extend to θ = 0. Geometrically SBn γ /OSBn γ is the rim of the spherical bowl SBn γ , namely the (n−1)-sphere centered at P = (0, 0, . . . , γ)T with radius √ 1− γ2; see Figure 3. We define ω := γ2 + (1− γ2)r2 = γ2 + (1− γ2)(q21 + q22 + · · ·+ q2n). (3.20) Then (3.14) can be rewritten as θ = γ2 + √ ω r2 + γ2 . (3.21) Given an approximation sequence (Qm)m∈N ⊂ R̃n to a unit vector U = (u1, · · · , un, 0) T , i.e. Qm := (q1m, q2m, . . . , qnm, 0)T , Um := (u1m, u2m, . . . , unm, 0)T = Qm ∥Qm∥ , rm := ∥Qm∥, Qm = rmUm = (rmu1m, rmu2m, . . . , rmunm, 0)T , lim m→∞ rm = ∞, lim m→∞ Um = U, Since Qm = (Qm, 0)T , (see (3.8)), Proposition 3.5 implies that each Qm is mapped to the point Zm in OSBn γ , where Zm := (θmrmu1m, θmrmu2m, . . . , θmrmunm, γ(1− θm))T θm := γ2 + √ ωm r2m + γ2 , ωm := γ2 + (1− γ2)r2m. Since γ is a fixed positive constant and since rm → ∞, we deduce that √ ωm ∼ √ 1− γ2rm, θm ∼ √ 1− γ2 rm , m → ∞ (3.22) which in turn implies Zm ∼ ( √ 1− γ2u1, √ 1− γ2u2, . . . , √ 1− γ2un, γ) T ∈ SBn γ /OSBn γ , m → ∞. Conversely, given Z = (x1, x2, . . . , xn, γ) T ∈ SBn γ /OSBn γ , observe that n∑ i=1 x2 i + γ2 = 1 ⇐⇒ √√√√ n∑ i=1 x2 i = √ 1− γ2. (3.23) For any sequence (Zm)m∈N ⊂ SBn,+ γ where Zm = (x1m, x2m, . . . xnm, xn+1m)T , and lim m→∞ Zm = Z, since Proposition 3.5 implies that each Zm is mapped to G−1(Zm) = γ γ − √ 1− ∑n i=1 x 2 im (x1m, x2m, . . . , xnm)T we apply (3.23) and find that lim m→∞ G−1(Zm) = lim m→∞ γ γ − √ 1− ∑n i=1 x 2 im (x1m, x2m, . . . , xnm)T = lim m→∞ γ √∑n i=1 x 2 im γ − √ 1− ∑n i=1 x 2 im (x1m, x2m, . . . , xnm)T√∑n i=1 x 2 im EJDE-2025/33 SPHERICAL COMPACTIFICATIONS 13 = γ √∑n i=1 x 2 i γ − √ 1− ∑n i=1 x 2 i (x1, x2, . . . , xn) T√∑n i=1 x 2 i = ∞ (x1, x2, . . . , xn) T√ 1− γ2 . In summary, the preceding calculations justify the construction of the following bijection be- tween SBn γ and UERn. Proposition 3.6. Given P = (0, 0, . . . , γ)T ∈ Rn+1, where 0 < γ < 1, let G : Rn → OSBn γ be the continuous bijection defined in Proposition 3.5. Define Ĝ : UERn → SBn γ as Ĝ(Q) = { G(Q), Q ∈ Rn ( √ 1− γ2u1, √ 1− γ2u2, . . . , √ 1− γ2un, γ) T , Q = ∞U ∈ IDn. (3.24) Then Ĝ is a set-theoretic bijection with inverse Ĝ−1 : SBn γ → UERn defined via Ĝ−1(Z) = G−1(Z), Z ∈ OSBn γ ∞ ( x1√ 1−γ2 , x2√ 1−γ2 , . . . , xn√ 1−γ2 )T , Z ∈ SBn γ /OSBn γ . (3.25) Definition 3.7. The map Ĝ : UERn → SBn γ of Proposition 3.6 is the spherical compactification of UERn associated with the parameter γ. By using the chordal distance between any two points Z, Ẑ ∈ SBn γ , i.e. the standard Euclidean distance in Rn+1, we can induce a complete metric on UERn and turn the bijection of Proposition 3.6 into a continuous bijection. Theorem 3.8. Let Ĝ be as defined in Proposition 3.6. Let ∥ ∥ denote the Euclidean norm in Rn+1. Then UERn is a complete metric space with respect to the chordal metric χ : UERn×UERn → R+, where χ(Q, Q̂) = ∥Ĝ(Q)− Ĝ(Q̂)∥. (3.26) In particular, if Q = (q1, q2, . . . , qn) T ∈ Rn, Q̂ = (q̂1, q̂2, . . . , q̂n) T ∈ Rn, Z = Ĝ(Q), and Ẑ = Ĝ(Q̂), Equation (3.26) becomes χ2(Q, Q̂) = ∥Z − Ẑ∥2 = 2− 2 (γ2 + √ ω r2 + γ2 )(γ2 + √ ω̂ r̂2 + γ2 )( n∑ i=1 qiq̂i + γ2 ) − 2 [ γ2 − γ2 (γ2 + √ ω r2 + γ2 ) − γ2 (γ2 + √ ω̂ r̂2 + γ2 )] , (3.27) where r2 := QTQ, r̂2 := Q̂T Q̂, ω := γ2 + (1− γ2)r2, ω̂ := γ2 + (1− γ2)r̂2; see Figure 4. If Q = (q1, q2, . . . , qn) T ∈ Rn and Q̂ = ∞Û , with Û = (û1, û2, . . . , ûn) T ∈ Sn−1, Equation (3.26) becomes χ2(Q, Q̂) = 1− γ2 + (γ2 + √ ω r2 + γ2 )2 (r2 + γ2)− 2 (γ2 + √ ω r2 + γ2 )√ 1− γ2 n∑ i=1 qiûi; (3.28) see Figure 5. If Q = ∞U , with U = (u1, u2, . . . , un) T ∈ Sn−1, and Q̂ = ∞Û , with Û = (û1, û2, . . . , ûn) T ∈ Sn−1, then (3.26) becomes χ2(Q, Q̂) = 2(1− γ2) [ 1− n∑ i=1 uiûi ] ; (3.29) see Figure 6. 14 H. GINGOLD, J. QUAINTANCE EJDE-2025/33 P = (0,0,γ ) (0,0,-1) (0,0,1) (1,0,0) (0,1,0) ) ) ) ) ) ) ) )P =P = (0,0,γ )γ ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )γγ )γ (0,0, (0,0, ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) 2(x , x , x ) 1 3 Z = (q , q , 0)Q = ^ 2 (x , x , x ) 1 3 ^ ^ ^ ^ ^ Z = ^ 1 2 = (Q, 0) (q , q , 0)Q = 1 2 = (Q, 0) ^ ^ P = (0,0,γ ) Figure 4. The distance between Q and Q̂ is given by the chordal distance (green dashed line) between Z and Ẑ in SB2 γ . P = (0,0,γ ) (0,0,-1) (0,0,1) (1,0,0) (0,1,0) ) ) ) ) ) )P =P = (0,0,γ )γ ) ) ) ) ) ) ) ) )γγ )γ (0,0, (0,0, ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ( u , u , γ ) 1 2 ^ ^ ^ ^ Z = ODDD1 - γ2 U = (u , u , 0)1 2 ^ ODDD1 - γ2 NU ^ ^ (q , q , 0)Q = 1 2 = (Q, 0) Z = 2 (x , x , x ) 1 3 P = (0,0,γ ) Figure 5. The distance between Q and Q̂ = ∞Û is given by the chordal distance (green dashed line) between Z and Ẑ in SB2 γ . Proof. Since χ is derived from the Euclidean norm of Rn+1, it is easy to see that χ is indeed a distance function on UERn. We now derive Equation (3.27). Recall that Q = (q1, q2, . . . , qn) T and Q̂ = (q̂1, q̂2, . . . , q̂n) T . Then Proposition 3.6 implies that Ĝ(Q) = Z = (x1, x2, . . . , xn+1) = (θq1, θq2, . . . , θqn, (1− θ)γ)T , θ = γ2 + √ ω γ2 + r2 Ĝ(Q̂) = Ẑ = (x̂1, x̂2, . . . , x̂n+1) = (θ̂q̂1, θ̂q̂2, . . . , θ̂q̂n, (1− θ̂)γ)T , θ̂ = γ2 + √ ω̂ γ2 + r̂2 . The chordal distance between Ĝ(Q) and Ĝ(Q̂) becomes ∥Z − Ẑ∥2 = (x1 − x̂1) 2 + (x2 − x̂2) 2 + (x3 − x̂3) 2 + · · ·+ (xn+1 − x̂n+1) 2 = 2− 2[x1x̂1 + x2x̂2 + x3x̂3 + · · ·+ xn+1x̂n+1], since ∥Z∥ = ∥Ẑ∥ = 1 = 2− 2[θθ̂q1q̂1 + θθ̂q2q̂2 + · · ·+ θθ̂qnq̂n + γ2(1− θ)(1− θ̂)] EJDE-2025/33 SPHERICAL COMPACTIFICATIONS 15 P = (0,0,γ ) (0,0,-1) (0,0,1) (1,0,0) (0,1,0) ) ) ) ) ) ) ) ) ) ) ) ) ) (0,0, (0,0, ) ) ) )P =P = (0,0,γ )γ ) ) ) ) ) ) ) ) ) ) ) ) ) )γγ )γ (0,0, (0,0, ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ( u , u , γ ) 1 2 Z = ^ ^ ^ ^ Z = ODDD1 - γ2 U = (u , u , 0)1 2 ^ ODDD1 - γ2 NU ^ ^ ( u , u , γ )ODDD1 - γ2 ODDD1 - γ2 1 2 U = (u , u , 0) 1 2 NU P = (0,0,γ ) Figure 6. The distance between Q = ∞U and Q̂ = ∞Û is given by the chordal distance (green line) between Z and Ẑ in SB2 γ . = 2− 2 (γ2 + √ ω γ2 + r2 )(γ2 + √ ω̂ γ2 + r̂2 ) [q1q̂1 + qq̂2 + · · ·+ qnq̂n] − 2 ( 1− γ2 + √ ω γ2 + r2 )( 1− γ2 + √ ω̂ γ2 + r̂2 ) γ2, which upon expansion of the third term is seen to be identical to Equation (3.27). To obtain (3.28) with Q = (q1, q2, . . . , qn) T and Q̂ = ∞Û with Û = (û1, û2, . . . , ûn) T , Proposi- tion 3.6 implies that Ĝ(Q) = Z = (θq1, θq2, . . . , θqn, (1− θ)γ)T , θ = γ2 + √ ω γ2 + r2 Ĝ(Q̂) = Ẑ = ( √ 1− γ2û1, √ 1− γ2û2, . . . , √ 1− γ2ûn, γ) T , and we find that (recall that ∥Û∥ = 1) ∥Z − Ẑ∥2 = n∑ i=1 ( √ 1− γ2ûi − θqi) 2 + (γ − (1− θ)γ)2 = (1− γ2) + θ2(q21 + q22 + · · ·+ q2n + γ2)− 2θ √ 1− γ2[q1û1 + q2û2 + · · ·+ qnûn], which, since r2 = ∑n i=1 q 2 i and θ = γ2+ √ ω γ2+r2 , is identical to (3.28). To verify (3.29), let Q = ∞U with U = (u1, u2, . . . , un) T , and let Q̂ = ∞Û with U = (û1, û2, . . . , ûn) T . Proposition 3.6 implies that Ĝ(Q) = Z = ( √ 1− γ2u1, √ 1− γ2u2, . . . , √ 1− γ2un, γ) T , Ĝ(Q̂) = Ẑ = ( √ 1− γ2û1, √ 1− γ2û2, . . . , √ 1− γ2ûn, γ) T . Then ∥Z − Ẑ2∥ = n∑ i=1 ( √ 1− γ2ui − √ 1− γ2ûi) 2 = (1− γ2) n∑ i=1 (u2 i − 2uiûi + û2 i ), which is equivalent to (3.29) since ∥U∥ = ∥Û∥ = 1. With the χ metric placed on UERn, and with SBn γ given the induced Euclidean metric from Rn+1, Proposition 3.6 defines a homeomorphism between SBn γ and UERn. Since SBn γ is a compact subspace of Rn+1, it is a complete metric space with respect to the subspace topology, which through the homeomorphism of Proposition 3.6, implies that UERn is also a complete metric space. □ Remark 3.9. Theorem 3.8 turns UERn into a complete metric space, but not a normed vector space. This is not surprising since Sn is not a vector space, but an n-manifold in Rn+1. The fact 16 H. GINGOLD, J. QUAINTANCE EJDE-2025/33 that UERn is not a vector space is also reflected by the fact that ∞U +∞Û is a set of elements in IDn rather than one element. This brings to the fore the distinction between the rules for calculating the scalar quantity ∞+∞ = ∞ and the rules for calculating ∞U +∞Û . For applications to differential equations, it is useful to work with an alternative version of the inverse provided by Proposition 3.6 in which we ignore the last coordinate xn+1 and per- pendicularly project SBn,+ γ onto the interior of the annulus An√ 1−γ2 ⊂ Rn, embeded into R̃n, where An√ 1−γ2 := {X ∈ Rn : √ 1− γ2 ≤ ∥X∥ ≤ 1}; (3.30) see Figure 7. Recall that Bn := {X ∈ Rn : 0 ≤ ∥X∥ ≤ 1}, ◦ Bn := {X ∈ Rn : 0 ≤ ∥X∥ < 1}. radius = (1 - radius = (1 - radius = (1 - radius = (1 - radius = (1 - radius = (1 - radius = (1 - radius = (1 - radius = (1 - radius = (1 - radius = (1 - radius = (1 - radius = (1 - radius = (1 - radius = (1 - radius = (1 - radius = (1 - radius = (1 - radius = (1 - radius = (1 - radius = (1 - radius = (1 - radius = (1 - radius = (1 - radius = (1 - radius = (1 - radius = (1 - radius = (1 - radius = (1 - radius = (1 - radius = (1 - radius = (1 - radius = (1 - radius = (1 - radius = (1 - radius = (1 - radius = (1 - radius = (1 - radius = (1 - radius = (1 - γ )γ2 ) ) 1/2 (0,0,1) (1,0,0) (0,1,0) (0,0,0) P = (0,0,γ ) Figure 7. “top half” of SB2 γ perpendicularly projected onto the “base” annulus in the xy-plane. Thus instead of using Z = (Z̃, xn+1) ∈ SBn,+ γ as part of the domain of Ĝ−1 (respectively G−1), we instead use Z̃ ∈ An√ 1−γ2 and define the following alternative inverse mapping. Proposition 3.10. Define K : An√ 1−γ2 → UERn/ ◦ Bn as K(Z̃) =  γ γ− √ 1− ∑n i=1 x2 i Z̃, Z̃ = (x1, x2, . . . , xn) T , √ 1− γ2 < ∥Z̃∥ ≤ 1 ∞ Z̃√ 1−γ2 , ∥Z̃∥ = √ 1− γ2. (3.31) Then K is a bijection between An√ 1−γ2 and UERn/ ◦ Bn, which is also referred to as the spherical compactification associated with the parameter γ. Observe that IDn is the image of {X ∈ An√ 1−γ2 : ∥X∥ = √ 1− γ2}, i.e. the inner boundary of the “annulus”; see Figure 8. Moreover, since there is a continuous bijection between An√ 1−γ2 and {(x1, x2, · · · , xn+1) T ∈ SBn γ : xn+1 ≥ 0}, namely (x1, x2, . . . , xn) T → ( x1, x2, . . . , xn, √√√√1− n∑ i=1 x2 i )T , we may transfer the chordal distance of SBn γ onto An√ 1−γ2 and use this transferred chordal distance as the metric of UERn/ ◦ Bn; see [30, Chapter 6]. EJDE-2025/33 SPHERICAL COMPACTIFICATIONS 17 K radius = (1 - γ ) 2 1/2 (1,0) (1,0) (0,1) (0,1) Figure 8. Annulus A2√ 1−γ2 is mapped to UER2/ ◦ B2 via an “inversion” over the black S1. In a similar manner, by ignoring the last coordinate xn+1, we perpendicularly project SBn,− γ onto ◦ Bn; see Figure 9. Proposition 3.11. Define K̂ : Bn → Bn as K̂(Z̃) = γ γ + √ 1− ∑n i=1 x 2 i Z̃, Z̃ = (x1, x2, . . . , xn) T , 0 ≤ ∥Z̃∥ ≤ 1. (3.32) Then K̂ is a bijection of Bn onto itself. By composing with the map (x1, x2, . . . , xn) T → ( x1, x2, . . . , xn,− √√√√1− n∑ i=1 x2 i )T , K̂ can also be considered as a bijection between SBn,− γ and ◦ Bn. P = (0,0, γ) (0,0,1) (1,0,0) (0,1,0) Figure 9. “lower half” of SB2 γ perpendicularly projected onto the unit disk in the xy-plane. 18 H. GINGOLD, J. QUAINTANCE EJDE-2025/33 4. Spherical Compactification of Differential Equations We now discuss how to apply the spherical compactification bijections to first order vector valued differential equations. Proposition 3.5, when combined with Propositions 3.10 and 3.11, implies that Z̃ = θQ ⇐⇒ Q = θ−1Z̃, θ = γ2 + √ γ2 + (1− γ2)QTQ γ2 +QTQ = 1∓ √ 1− Z̃T Z̃ γ , (4.1) where Z̃ ∈ Rn such that Z = (Z̃, xn+1) T ∈ OSBn γ , and Q ∈ Rn. Recall that θ > 0, so we refer to θ as a dilation factor. For the context of differential equations, we assign Q =⇒ w(t), and Z̃ =⇒ z(t). (4.2) With the conventions of (4.2), the formulas of (4.1) are succinctly written as z(t) = θ(t)w(t) ⇐⇒ w(t) = θ−1(t)z(t), θ(t) = γ2 + √ β γ2 + r2 = 1∓ √ 1−R2 γ , (4.3) where r(t) = ∥w(t)∥ = √ wT (t)w(t), β(t) = γ2 + (1− γ2)r2, R(t) = ∥z(t)∥ = √ zT (t)z(t). (4.4) Previously β was denoted as ω, but because w(t) so closely resembles ω, we decided to change the notation. Furthermore, to alleviate notation, we often write w(t) = w, z(t) = z, and θ(t) = θ, etc. Remark 4.1. Since θ∥w(t)∥ = ∥z(t)∥, when θ ̸= 0, we deduce that z(t) ∥z(t)∥ = θw(t) θ∥w(t)∥ = w(t) ∥w(t)∥ . (4.5) Equation (4.5) shows that (4.3) maps a unit vector of UERn (which is associated with w(t)) to a unit vector in the projection of SBn γ onto Rn (which is associated with z(t)). This is crucial when discussing the notion of critical points (constant solutions) to w(t) = F (w(t)) of the form ∞U , since (4.5) implies that lim t→∞ w(t) ∥w(t)∥ = lim t→∞ z(t) ∥z(t)∥ = U, (4.6) given that the limit exists. Suppose we have a first order vector valued differential equation of the form w′(t) = F (w(t)), where w(t) ∈ UERn. Assume that θ ̸= 0. We use w(t) = θ−1z(t) and convert w′(t) = F (w(t)) into z′(t) = H(z(t)). We want to investigate the correspondence, if any, between finite critical points wcp of w′ = F (w) and finite critical points zcp of z′ = H(z). Since dθ dt = ± zT z′ γ √ 1− zT z , (4.7) and since w(t) = θ−1(t)z(t), we discover that w′(t) = −θ−2 dθ dt z + θ−1z′ = −θ−2 [ ± zT z′ γ √ 1− zT z ] z + θ−1z′ = −θ−2 [ ± zzT γ √ 1− zT z ] z′ + θ−1z′ = θ−1 [ In ∓ θ−1zzT γ √ 1− zT z ] z′. (4.8) As long as ∥z∥ ≠ 1, Equation (4.8) implies that z′ = θ [ In ∓ θ−1zzT γ √ 1− zT z ]−1 w′. (4.9) EJDE-2025/33 SPHERICAL COMPACTIFICATIONS 19 It is now a matter of calculating [In +∆]−1 where ∆ := ∓ θ−1zzT γ √ 1− zT z . (4.10) We will assume that (In +∆)−1 = In + µ∆, which in turn implies that In = (In +∆)(In +∆)−1 = (In +∆)(In + µ∆) = In +∆+ µ∆+ µ∆2. (4.11) Since R2 := zT z, we find that ∆2 = θ−2 γ2(1−R2) z(zT z)zT = θ−2R2 γ2(1−R2) zzT . Then (4.11) becomes 0 = ∆+ µ∆+ θ−2R2 γ2(1−R2) µzzT = [∓γθ−1 √ 1−R2 ∓ γθ−1 √ 1−R2µ+ µθ−2R2 γ2(1−R2) ] zzT . Now we set the numerator to zero and solve for µ as µ = ±γ √ 1−R2 θ−1R2 ∓ γ √ 1−R2 ̸= 0. (4.12) By using (4.12), we can rewrite (4.9) as z′ = θ [ In − θ−1zzT θ−1zT z ∓ γ √ 1− zT z ] w′. (4.13) As long as ∥z∥ ≠ 1 and θ > 0, (θ is always finite by Part (iii) of Proposition 4.6), the matrix In − θ−1zzT /(θ−1zT z ∓ γ √ 1− zT z) is invertible. Because θ = 1 ∓ γ−1 √ 1− zT z, the fact that ∥z∥ ̸= 1 is equivalent to the fact that θ ̸= 1. Since ∥z∥ = 1 if and only if θ = 1 if and only if z = θw = w, we deduce the one-to-one correspondence between finite critical points wcp of w′ = F (w) and finite critical points zcp of z′ = H(z) such that ∥wcp∥, ∥zcp∥ ≠ 1. The question remains what happens to (4.13) if ∥z∥ = 1. This requires letting zT z → 1 in (4.13) to obtain z′ = [In − zzT ]w′. (4.14) From (4.14) we deduce that a critical point wcp of w′ = F (w) is mapped to a critical point zcp of z′ = H(w). However, there could be critical points of z′ = F (z) which are eigenvectors of the noninvertible matrix In − zzT . In summary we have proven the following theorem. To rewrite (4.13) we make the following definition. Definition 4.2. Let w(t), w′(t), z(t), z′(t) ∈ C[t0,∞), with w(t), w′(t) ∈ Rn, with z(t), z′(t) ∈ Bn, and where w(t) and z(t) are related via (4.3). Assume that w′(t) = F (w) and z′(t) = Hi(z), i ∈ {1, 2}, where F : Rn → Rn, H1 : An√ 1−γ2 → Rn, H2 : Bn → Rn, (4.15) are three functions which satisfy the following conditions. (i) F (w) is continuous in some compact connected set DF ⊆ Rn. (ii) H1(z) is continuous in some compact connected set DH1 ⊆ An√ 1−γ2 . (iii) H2(z) is continuous in some compact connected set DH2 ⊆ Bn. (iv) If ∥w∥ = 1 = ∥z∥, or equivalently if θ = 1, H1 ≡ H2, w ′(t) = F (w), and z′(t) = H1(z) = H2(z). (v) For ∥w∥ > 1, or equivalently for 0 < θ < 1, ( z, √ 1− ∥z∥2 )T ∈ SBn,+ γ , z′(t) = H1(z), and w′(t) = F (w). (vi) For ∥w∥ < 1, or equivalently for θ > 1, ( z,− √ 1− ∥z∥2 )T ∈ SBn,− γ , z′(t) = H2(z), and w′(t) = F (w). 20 H. GINGOLD, J. QUAINTANCE EJDE-2025/33 Theorem 4.3. Let F , H1, and H2 be given by Definition 4.2. If 0 < θ < 1, or equivalently if ∥w∥ > 1, Equation (4.8) becomes w′(t) = F (w) = θ−1 [ In − θ−1zzT γ √ 1− zT z ] H1(z). (4.16) Given a critical point zcp of H1(z), i.e. H1(zcp) = 0⃗, since θ−1 cp = [1 − √ 1− ∥zcp∥2/γ]−1 and w = θ−1z, Equation (4.16) implies that w′(t) = F (θ−1 cp zcp) = θ−1 cp [ In − θ−1 cp zcpz T cp γ √ 1− zTcpzcp ] H1(zcp) = 0⃗, i.e. θ−1 cp zcp is a constant solution of w′(t) = F (w). Since z = θw, Equation (4.13) becomes z′ = H1(z) = θ [ In − θwwT θwTw − γ √ 1− θ2wTw ] F (w). (4.17) Given a critical point wcp of F (w), i.e. F (wcp) = 0⃗, since θcp = [γ2 + √ γ2 + (1− γ2)∥wcp∥2]/[γ2 + ∥wcp∥2], Equation (4.17) implies that z′(t) = H1(θcpwcp) = θcp [ In − θcpwcpw T cp θwT cpwcp − γ √ 1− θ2cpw T cpwcp ] F (wcp) = 0⃗, i.e. θcpwcp is a constant solution of z′(t) = H1(z). Because the n×n matrices in (4.16) and (4.17) are invertible, there is a one-to-one correspon- dence between the finite critical points wcp of w′ = F (w) such that ∥wcp∥ > 1 and the finite critical points zcp of z′ = H1(z), where ∥zcp∥ < 1. If θ > 1, or equivalently if ∥w∥ < 1, Equation (4.8) becomes w′(t) = F (w) = θ−1 [ In + θ−1zzT γ √ 1− zT z ] H2(z). (4.18) Given a critical point zcp of H2(z), i.e. H2(zcp) = 0⃗, since θ−1 cp = [1 + √ 1− ∥zcp∥2/γ]−1 and w = θ−1z, Equation (4.18) implies that w′(t) = F (θ−1 cp zcp) = θ−1 cp [ In + θ−1 cp zcpz T cp γ √ 1− zTcpzcp ] H2(zcp) = 0⃗, i.e. θ−1 cp zcp is a constant solution of w′(t) = F (w). Since z = θw, Equation (4.13) becomes z′ = H2(z) = θ [ In − θwwT θwTw + γ √ 1− θ2wTw ] F (w). (4.19) Given a critical point wcp of F (w), i.e. F (wcp) = 0⃗, since θcp = [γ2 + √ γ2 + (1− γ2)∥wcp∥2]/[γ2 + ∥wcp∥2], Equation (4.19) implies that z′(t) = H2(θcpwcp) = θcp [ In − θcpwcpw T cp θwT cpwcp + γ √ 1− θ2cpw T cpwcp ] F (wcp) = 0⃗, i.e. θcpwcp is a constant solution of z′(t) = H2(z). Because the n×n matrices in (4.18) and (4.19) are invertible, there is a one-to-one correspon- dence between the finite critical points wcp of w′ = F (w) such that ∥wcp∥ < 1 and the finite critical points zcp of z′ = H2(z), where ∥zcp∥ < 1. EJDE-2025/33 SPHERICAL COMPACTIFICATIONS 21 If θ = 1, or equivalently if z = w with ∥z∥ = 1, Equation (4.14) becomes z′(t) = H1(z) ≡ H2(z) = [In − zzT ]F (w) = [In − wwT ]F (w), (4.20) and a finite critical point wcp of w′ = F (w) with ∥wcp∥ = 1 is mapped to a finite critical point zcp of z′ = H1(z) ≡ H2(z), where ∥zcp∥ = 1. There could be additional finite critical points of zcp of z′ = H1(z) ≡ H2(z) if F (w) is eigenvector of [In − zzT ] for eigenvalue 0. We obtain a refinement of Theorem 4.3 if we assume the critical points in question are attainable. The definition of attainability will be motivated by the following lemma. Lemma 4.4. Let h : R → R, let h′ : R → R, and assume that h(t), h′(t) ∈ C[t0,∞). Suppose that lim t→∞ h(t) = L1, and that lim t→∞ h′(t) = L2, (4.21) where |L1| < ∞ and |L2| < ∞. Then L2 = 0. Proof. This is a proof by contradiction. First assume that L2 > 0. The second limit of (4.21) implies there exists t1 ∈ [t0,∞) such that h′(t) ≥ L2/2 for all t ∈ [t1,∞). Then for t2 ∈ [t1,∞) h(t2) = h(t1) + ∫ t2 t1 h′(s) ds ≥ h(t1) + ∫ t2 t1 L2 2 ds = h(t1) + (t2 − t1) L2 2 . If we take the limit of the above inequalities, we find that limt2→∞ h(t2) = L1 ≥ ∞, which is a contradiction to the fact that |L1| < ∞. The case of L2 < 0 is left to the reader. □ Suppose we have a vector valued differential equation x′(t) = P (x) such that x(t) ∈ C[t0,∞). Furthermore assume that P (x) is continuous on some connected compact DP ⊆ Rn and there is some xcp ∈ DP such that limt→∞ x(t) = xcp. The continuity of P and x(t) implies that lim t→∞ x′(t) = lim t→∞ P (x(t)) = P ( lim t→∞ x(t) ) = P (xcp) < ∞. (4.22) Since limt→∞ x′(t) = P (xcp), a component by component application of Lemma 4.4 shows that limt→∞ x′(t) = 0⃗, which in turn implies that P (xcp) = 0⃗ for all t ∈ R. This phenomena is recorded in the following definition. Definition 4.5. Let x′(t) = P (x) be a first ordered vector valued differential equation taking values in Rn. Assume that x(t) ∈ C[t0,∞) and P (x) is continuous over DP ⊆ Rn, where DP is a compact connected set such that xcp ∈ DP . If limt→∞ x(t) = xcp, then xcp is a finite attainable critical point of x′(t) = P (x). Before we state our first refinement of Theorem 4.3, we need some properties of θ. Proposition 4.6. Let θ, β, and r be as defined in (4.3) and (4.4). Then dθ dr2 = (γ2 + r2)(1− γ2)− 2(γ2 + √ β) √ β 2 √ β(γ2 + r2)2 . (4.23) and θ satisfies the following properties: (i) θ = O(r−1) as r → ∞. (ii) The dilation factor θ is a monotone decreasing function of r2 > 0. (iii) 0 ≤ θ ≤ 1 + γ−1. Proof. Property (i) is a restatement of (3.22) with θm playing the role of θ. To obtain (4.23) we differentiate the first expression of θ provided by (4.3). Since 0 < γ < 1, we deduce that β > 0, and that the denominator of (4.23) is never 0. Thus dθ/dr2 is a continuous rational function in the variables r2, √ β, and γ whose sign is determined by (γ2 + r2)(1− γ2)− 2(γ2 + √ β) √ β, which upon expansion becomes (γ2 + r2)(1− γ2)− 2(γ2 + √ β) √ β = −γ2 − γ4 − (1− γ2)r2 − 2γ2 √ β < 0. Thus dθ/dr2 < 0, which in turn implies (ii). Since θ is monotone decreasing with respect to r2, the maximum value of θ(r2) is θ(0) = (γ2 + γ)/(γ2) = 1 + γ−1, while the minimum value of θ is limr2→∞ θ(r2) = 0. □ 22 H. GINGOLD, J. QUAINTANCE EJDE-2025/33 Proposition 4.6 will be used in the proof of the following proposition which shows that a finite attainable critical point wcp of w′(t) = F (w(t)) transforms into a finite attainable critical point zcp of z′(t) = H(z(t)). Proposition 4.7. Let F , H1, and H2 be given by Definition 4.2. Assume that limt→∞ w(t) = L ∈ DF , where ∥L∥ ≠ 1, i.e. L is a finite attainable critical point of w′ = F (w). If ∥L∥ > 1, assume that θ(L)L ∈ DH1 , while if ∥L∥ < 1, assume that θ(L)L ∈ DH2 . Then following conditions hold: (a) limt→∞ r(t) = ∥L∥, lim t→∞ θ(t) = γ2 + √ γ2 + (1− γ2)∥L∥2 γ2 + ∥L∥2 ≡ θ(L). (b) limt→∞ dθ dt = 0, limt→∞ dz dt = 0⃗. (c) If ∥L∥ > 1, then θ(L)L is a finite attainable critical point of z′(t) = H1(z). (d) If ∥L∥ < 1, then θ(L)L is a finite attainable critical point of z′(t) = H2(z). Proof. Part (a) follows from the continuity of r(t) and θ(t). The differentiability θ(t) follows from dθ dt = dθ dr2 dr2 dt . Since z(t), w(t), and θ(t) are differentiable over [t0,∞), we obtain dz dt = dθ dt w(t) + θ(t) dw dt . (4.24) If we take the limit of (4.24), since the paragraph before Definition 4.5 implies that limt→∞ w′(t) = 0⃗, we obtain lim t→∞ z′(t) = lim t→∞ ( θ(t) dw dt ) + lim t→∞ (dθ dt w(t) ) = L lim t→∞ dθ dt . (4.25) So it is now a matter of computing lim t→∞ dθ dt = lim t→∞ dθ dr2 lim t→∞ dr2 dt . (4.26) Equation (4.23), along with the continuity of r(t) and w(t), implies that lim t→∞ dθ dr2 = L1 < ∞, (4.27) where L1 := (γ2 + ∥L∥2)(1− γ2)− 2(γ2 + √ γ2 + (1− γ2)∥L∥2) √ γ2 + (1− γ2)∥L∥2 2 √ γ2 + (1− γ2)∥L∥2(γ2 + ∥L∥2)2 . Since dr2 dt = 2wTw′, we deduce that limt→∞ dr2 dt = 0. Substituting the above calculations into the right side of (4.26) gives us lim t→∞ dθ dt = L1 lim t→∞ dr2 dt = 0. (4.28) We substitute (4.28) into (4.25) and conclude that limt→∞ z′(t) = 0⃗. To prove (c) and (d) we use the continuity of z(t) and Hi(z), where i ∈ {1, 2}, along with (a) and (b). In particular, if ∥L∥ > 1, there exits t1 > 0 such that ∥w(t)∥ > 1 for all t ∈ [t1,∞). Then 0⃗ = lim t→∞ z′(t) = lim t→∞ H1(z(t)) = H1( lim t→∞ z(t)) = H1( lim t→∞ θ(t)w(t)) = H1 (γ2 + √ γ2 + (1− γ2)∥L∥2 γ2 + ∥L∥2 L ) . If ∥L∥ < 1, there exists t2 > 0 such that ∥w(t)∥ < 1 for all t ∈ [t2,∞). Then 0⃗ = lim t→∞ z′(t) = lim t→∞ H2(z(t)) = H2( lim t→∞ z(t)) = H2( lim t→∞ θ(t)w(t)) EJDE-2025/33 SPHERICAL COMPACTIFICATIONS 23 = H2 (γ2 + √ γ2 + (1− γ2)∥L∥2 γ2 + ∥L∥2 L ) . □ Remark 4.8. If ∥L∥ = 1, since H1(z) ≡ H2(z) when ∥z(t)∥ = 1, the last two limit calculations in the proof of Proposition (4.7) combine to show that 0⃗ = lim t→∞ z′(t) = lim t→∞ Hi(z(t)) = Hi( lim t→∞ z(t)) = Hi( lim t→∞ θ(t)w(t)) = H1(L) ≡ H2(L). Next we prove the converse of Proposition 4.7 and show that under mild conditions a finite attainable critical point of zcp of z′(t) = H(z(t)) transforms into a finite attainable critical point wcp of w′(t) = F (w(t)). Proposition 4.9. Let F , H1, and H2 be given by Definition 4.2. Assume that limt→∞ z(t) = C, where C ∈ Ån√ 1−γ2 , and that C ∈ DH1 ∪DH2 . This implies that lim t→∞ ( z(t), √ 1− ∥z(t)∥2 )T = ( C, √ 1− ∥C∥2 )T ∈ SBn,+ γ lim t→∞ ( z(t),− √ 1− ∥z(t)∥2 )T = ( C,− √ 1− ∥C∥2 )T ∈ SBn,− γ . (4.29) Furthermore assume that [1∓ γ−1 √ 1− ∥C∥2]−1C ∈ DF . Then the following conditions hold. (a) limt→∞ R(t) = ∥C∥, limt→∞ θ(t) = 1∓ √ 1−∥C∥2 γ . (b) limt→∞ dθ dt = 0, limt→∞ dz dt = 0⃗. (c) limt→∞ dw dt = 0⃗. (d) z′(t) = Hi(z) has a finite attainable critical point C, where i ∈ {1, 2}. (e) w′(t) = F (w) has finite attainable critical points θ−1(C)C = [ 1∓ √ 1−∥C∥2 γ ]−1 C. Proof. Observe that limt→∞ R(t) = ∥C∥ follows from the continuity of R(t). Also the continuity of z(t), z′(t), and Hi(z) for i ∈ {1, 2} implies that lim t→∞ z′(t) = lim t→∞ Hi(z(t)) = Hi ( lim t→∞ z(t) ) = Hi(C) < ∞. (4.30) Then Lemma 4.4 implies that limt→∞ z′(t) = 0⃗, and that Hi(C) = 0, i.e. C is a constant solution of Hi(z(t)) = z′(t). Observe that (4.30) is independent of whether ∥C∥ < 1 or ∥C∥ = 1. To prove that limt→∞ dθ dt = 0, that limt→∞ θ(t) = 1 ∓ √ 1−∥C∥2 γ , and to verify (e), we have to analyze the location of the preimage of C on SBn γ . Case 1: Take C ∈ DH1 , namely that (C, √ 1− ∥C∥2)T ∈ SBn,+ γ . Then θ(t) = 1 − √ 1−R2 γ . As long as R2 ̸= 1 − γ2, θ ̸= 0, and θ−1 is well defined. This is not a problem since C ∈ Ån√ 1−γ2 . Hence the continuity of R(t) implies that lim t→∞ θ(t) = 1− √ 1− ∥C∥2 γ ̸= 0, (4.31) lim t→∞ θ(t)−1 = [ 1− √ 1− ∥C∥2 γ ]−1 := L2, |L2| < ∞, (4.32) lim t→∞ θ(t)−2 = [ 1− √ 1− ∥C∥2 γ ]−2 := L3, |L3| < ∞. (4.33) Since (C, √ 1− ∥C∥2)T ∈ SBn,+ γ , R2 ̸= 1, and we may differentiate θ(t) as in (4.7) to find that lim t→∞ dθ dt = lim t→∞ z(t)T z′(t) γ √ 1− z(t)T z(t) = 0. (4.34) 24 H. GINGOLD, J. QUAINTANCE EJDE-2025/33 Next observe that w(t) = θ−1(t)z(t) is differentiable with derivative dw dt = −θ−2 dθ dt z(t) + θ−1 dz dt . (4.35) If we take the limit of (4.35) and use (4.32), (4.33), and (4.34), along with the fact that limt→∞ z′(t) = 0⃗, we obtain Part (c). Since (C, √ 1− ∥C∥2)T ∈ SBn,+ γ , there is t1 > 0 such that 0 < θ(t) < 1 for t ∈ [t1,∞). The continuity of F shows that 0⃗ = lim t→∞ w′(t) = lim t→∞ F (w(t)) = F ( lim t→∞ w(t) ) = F ( lim t→∞ θ−1(t)z(t) ) = F ([ 1− √ 1− ∥C∥2 γ ]−1 C ) . Hence w′(t) = F (w) has a constant solution [ 1− √ 1−∥C∥2 γ ]−1 C for all t ∈ R. Case 2: Take C ∈ DH2 , namely that (C,− √ 1− ∥C∥2)T ∈ SBn,− γ . Then θ(t) = 1 + √ 1−R2 γ ̸= 0. Since θ−1(t) is well defined, and with minor changes of signs, the proof of Case 1 is applicable. Since (C,− √ 1− ∥C∥2)T ∈ SBn,− γ , there exists t2 > 0 such that θ(t) > 1 for t ∈ [t2,∞), which when combined with the continuity of w and F implies that 0⃗ = lim t→∞ w′(t) = lim t→∞ F (w(t)) = F ( lim t→∞ w(t) ) = F ( lim t→∞ θ−1(t)z(t) ) = F ([ 1 + √ 1− ∥C∥2 γ ]−1 C ) . Hence w′(t) = F (w) has a constant solution [1 + √ 1−∥C∥2 γ ]−1C. □ The proof of Case 2 of Proposition 4.9 also proves the following proposition. Proposition 4.10. Let F , H1, and H2 be given by Definition 4.2. Assume that limt→∞ z(t) = C, where C ∈ ◦ B n√ 1−γ2 = {z : |z| ≤ √ 1− γ2}, and that C ∈ DH2 . This implies that lim t→∞ ( z(t),− √ 1− ∥z(t)∥2 )T = ( C,− √ 1− ∥C∥2 )T ∈ SBn,− γ . (4.36) Furthermore assume that [1 + γ−1 √ 1− ∥C∥2]−1C ∈ DF . Then the following conditions hold. (a) limt→∞ R(t) = ∥C∥, limt→∞ θ(t) = 1 + √ 1−∥C∥2 γ . (b) limt→∞ dθ dt = 0, limt→∞ dz dt = 0⃗. (c) limt→∞ dw dt = 0⃗. (d) z′(t) = H2(z) has a finite attainable critical point C (e) w′(t) = F (w) has a finite attainable critical point θ−1(C)C = [ 1 + √ 1−∥C∥2 γ ]−1 C. Proposition 4.7, when combined with Propositions 4.9 and 4.10, provides the proof of Theorem 4.11. EJDE-2025/33 SPHERICAL COMPACTIFICATIONS 25 Theorem 4.11. Let F , H1, and H2 be given by Definition 4.2. There exists a one-to-one corre- spondence between finite attainable critical points wcp of w′ = F (w), where ∥wcp∥ > 1 and finite attainable critical points of zcp of z′(t) = H1(z), where ∥zcp∥ < 1. There also exists a one-to-one correspondence between finite attainable critical points wcp of w′ = F (w), where ∥wcp∥ < 1 and finite attainable critical points of zcp of z′(t) = H2(z), where ∥zcp∥ < 1. 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Harry Gingold West Virginia University, Morgantown, WV, USA Email address: gingold@math.wvu.edu Jocelyn Quaintance University of Pennsylvania, Philadelphia, PA, USA Email address: jocelynq@seas.upenn.edu 1. Introduction 2. Algebraic structure of the ultra extended Rn 3. Spherical compactification of UERn 4. Spherical Compactification of Differential Equations References