# Inseparable

An inseparable assembly is new to Creo 8. It allows you to manage multi-component assemblies more efficiently as a single file. You might have thousands of parts and subassemblies in your model. Without the support of inseparable assemblies, your assembly could become cluttered and slow.

## Inseparable

Quantum mechanics allows the occurrence of events without having any definite causal order. We show that the application of two different thermal channels in causally inseparable order can enhance the potential to extract work, in contrast to any of their definite order of occurrence. Surprisingly, this enhancement is possible even without assigning any thermodynamic resource value to the input qubits. Further, we provide a nontrivial example of causal enhancement using nonunital pin maps, the realization of which, in the superposition of path framework, has not been clearly known. Hence, our result may be a potential candidate to accentuate the difference between superposition of time and superposition of paths frameworks.

The channel parameter is chosen as p=q=0.8 and input state probability r=0.5+0.01s, as mentioned earlier. The dashed blue curves stand for the causally inseparable order of occurrence, whereas the solid red curves stand for the causally separable cases. (a) Fixed bath scenario: The final free energy with respect to the input state parameter is plotted using a resourceful controller qubit. (b) Varying bath scenario: The free-energy difference between the final output and the initial state is plotted with the same controller.

The channel parameter is chosen as p=q=0.8 and input state probability r=0.5+0.01s, as mentioned earlier. (a) Fixed bath scenario: The solid blue straight line at the bottom denotes the free energy of the final output under the causally separable combination of the channels NPF and NGAD, whereas the dashed yellow line stands for the final free energy under a causally inseparable combination of these two channels using the free thermal state as the controlling qubit. (b) Varying bath scenario: The solid red curve denotes the free-energy difference between the final output (under the causally separable combination of the channels) and the initial state. The dashed blue line stands for the same under a causally inseparable combination of these two channels with a thermal controller.

Thermodynamic advancement using resource in the controller qubit. The yellow (lower) region denotes the variation of the final free energy under a causally separable order of occurrence for NGAD and NPF, and the blue (upper) region stands for their inseparable causal combination. The phase-flip parameter q is chosen as 0.3. In (a), the final free energy is calculated with respect to the bath fixed by the NGAD channel parameter p, whereas in (b), the same is calculated by assuming that the receiver has an access to the same bath as that of the input qubit.

Thermodynamic advancement using resource-free controller qubit. The phase-flip parameter q is chosen, as earlier. (a) The variation of the final free energy for a causally definite sequence of these two channels is depicted by the yellow (lower) region, whereas the blue (upper) region stands for a causally inseparable order of occurrence. (b) Here, the green (lighter) region depicts the causally inseparable case and the red (darker) region depicts the causally separable case.

Lemma 53.13.2. Let $k$ be a field of characteristic $p > 0$. Let $f : X \to Y$ be a nonconstant morphism of proper nonsingular curves over $k$. If the extension $k(X)/k(Y)$ of function fields is purely inseparable, then there exists a factorization

Proof. This follows from Theorem 53.2.6 and the fact that a finite purely inseparable extension of fields can always be gotten as a sequence of (inseparable) extensions of degree $p$, see Fields, Lemma 9.14.5. $\square$

Lemma 53.13.3. Let $k$ be a field of characteristic $p > 0$. Let $f : X \to Y$ be a nonconstant morphism of proper nonsingular curves over $k$. If $X$ is smooth and $k(Y) \subset k(X)$ is inseparable of degree $p$, then there is a unique isomorphism $Y = X^(p)$ such that $f$ is $F_X/k$.

Lemma 53.13.4. Let $k$ be a field of characteristic $p > 0$. Let $f : X \to Y$ be a nonconstant morphism of proper nonsingular curves over $k$. If $X$ is smooth and $k(Y) \subset k(X)$ is purely inseparable, then there is a unique $n \geq 0$ and a unique isomorphism $Y = X^(p^ n)$ such that $f$ is the $n$-fold relative Frobenius of $X/k$.

Example 53.13.6. This example will show that the genus can change under a purely inseparable morphism of nonsingular projective curves. Let $k$ be a field of characteristic $3$. Assume there exists an element $a \in k$ which is not a $3$rd power. For example $k = \mathbfF_3(a)$ would work. Let $X$ be the plane curve with homogeneous equation

as in Section 53.9. On the affine piece $D_+(T_0)$ using coordinates $x = T_1/T_0$ and $y = T_2/T_0$ we obtain $x^2 - y^3 + a = 0$ which defines a nonsingular affine curve. Moreover, the point at infinity $(0 : 1: 0)$ is a smooth point. Hence $X$ is a nonsingular projective curve of genus $1$ (Lemma 53.9.3). On the other hand, consider the morphism $f : X \to \mathbfP^1_ k$ which on $D_+(T_0)$ sends $(x, y)$ to $x \in \mathbfA^1_ k \subset \mathbfP^1_ k$. Then $f$ is a morphism of proper nonsingular curves over $k$ inducing an inseparable function field extension of degree $p = 3$ but the genus of $X$ is $1$ and the genus of $\mathbfP^1_ k$ is $0$.

Proof. By Fields, Lemma 9.14.6 there is a subextension $k(X)/E/k(Y)$ such that $k(X)/E$ is purely inseparable and $E/k(Y)$ is separable. By Theorem 53.2.6 this corresponds to a factorization $X \to Z \to Y$ of $f$ with $Z$ a nonsingular proper curve. Apply Lemma 53.13.4 to the morphism $X \to Z$ to conclude. $\square$

Simply defined, the doctrine of inseparable operations affirms that the triune persons act as a single agent externally, while internally their operations are divided.[2] Easy to say, harder to understand, even harder to teach and preach! Having taught this doctrine to graduate students, with particular emphasis during the last ten years, it is one of the most counter-intuitive doctrines they will have encountered in seminary. This is evidence for the domination of a functional tritheism in our churches, sometimes generated by a social understanding of the Trinity. By a functional tritheism I mean the belief that the different divine persons do their own thing, each having their own role, and each being responsible for certain effects. Most crudely, the Father alone is often thought to create, while only the Son redeems and only the Spirit sanctifies or perfects. Or, in what is one of the most damaging caricatures of all, the stern Father awaits in heaven for the Son to complete his mission, upon which he acts, restoring fellowship with sinners and sending the Holy Spirit. It gets worse when we begin to reflect on the cross itself: either the Father turns his eyes away from the Son, or the Father unilaterally acts to punish the Son, or the Father breaks relations with the Son, etc. These images populate our sermons, they have penetrated our hymnody, and they shape our collective consciousness.

The illustration of the magnet thus allows us to respond to a major dogmatic objection to inseparable operations: the objection from the incarnation of the Son alone. Just as the needle is drawn by the magnet as a whole, so is the incarnation the work of the whole Trinity; and just as the needle attaches distinctly to one of the poles, so the human nature of Christ is united distinctly with the Son of God. And just as the circle is but the shadow left in Flatland of the passing sphere, so the human nature of Christ is not to be simplistically confused with the Son, or with the whole Trinity. It is precisely God the Son, incarnate in human form. The persons of the Trinity are not separable parts of God but are defined and constituted by their relations with each other within the divine substance. Click To Tweet

The magnet can be of further use in helping us understand how the doctrine of inseparable operations naturally follows from the doctrine of divine simplicity and aseity. Just as we must distinguish between the inherent magnetism of the magnet and the action of the magnet upon needles, so we distinguish between the ad intra operations (the eternal generation of the Son from the Father, and the eternal procession of the Spirit from the Father and the Son) and the ad extra operations of the Trinity. Like the magnetic poles, which are functions of the magnetic directionality, the persons of the Trinity are not separable parts of God but are defined and constituted by their relations with each other within the divine substance. The Son proceeds as the self-knowledge of the Father, and the Spirit proceeds as the love between the Father and the Son. Because the persons do not have their own substance, they have no distinct principle of operation (which is what a substance is). The only other way in which they might have their own operations is within the unity of the substance. But these operations ad intra cannot involve any external reality, for in that case the persons (and God himself) would be constituted by and thus depend on something created for their identity, which would be an absurdity. 041b061a72