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+arXiv:2301.01909v1  [math-ph]  5 Jan 2023
+Solid-solid phase transitions in the ‘near-liquid’ limit
+Yury Grabovsky∗
+Lev Truskinovsky†
+January 6, 2023
+Abstract
+In this paper, dedicated to the memory of J. Ericksen, we address the fundamental
+difference between solid-solid and liquid-liquid phase transitions while remaining within
+the Ericksen’s nonlinear elasticity paradigm. To this end we assume that rigidity is
+weak and explore the nature of solid-solid phase transitions in a ‘near-liquid’ limit. In
+the language of calculus of variations we probe limits of quasiconvexity in an ’almost
+liquid’ solid by comparing the thresholds for cooperative (laminate based) and non-
+cooperative (inclusion based) nucleation. We consider a 2D problem and work with a
+prototypical two-phase Hadamard material. Using these two types of nucleation tests
+we obtain for this material surprisingly tight two-sided bounds on the elastic binodal
+without computing the quasi-convex envelope.
+1
+Introduction
+In 1975 J. Ericksen posed the problem of equilibrium for solids undergoing first order phase
+transitions in the framework of nonlinear elasticity theory. In this way he effectively reformu-
+lated the classical problem of physics into a problem of vectorial calculus of variations. The
+contemporaneous physical theory viewed non-hydrostatically stressed solids as metastable
+and therefore did not distinguish between solid-solid and liquid-liquid phase transitions. J.
+Ericksen realized that at normal conditions the assumption of complete relaxation of non-
+hydrostatic stresses is impractical and his pioneering research program of studying materials
+with non-rank-one convex energies revolutionized elasticity theory. The goal of this paper is
+to elucidate the difference between solid-solid and liquid-liquid phase transitions within the
+Ericksen’s nonlinear elasticity paradigm.
+From the perspective of elasticity theory, the main difference between liquids and solids
+is that liquids do not resist shear [6, 10]. This degeneracy in the elastic constitutive structure
+of liquids is responsible for their peculiar behavior during first order phase transitions vis a
+vis the behavior of solids, characterized by finite rigidity [17]. While in both cases reaching
+phase equilibrium usually leads to the formation of phase mixtures, in the case of solids
+∗Department of Mathematics, Temple University, Philadelphia, PA 19122, USA
+†PMMH, CNRS – UMR 7636, ESPCI, PSL, 75005 Paris, France
+1
+
+the knowledge of phase fractions carries considerably more information about the geometry
+of the resulting microstructure than in the case of liquids. More specifically, if the phase
+organization in liquid phase transitions is largely controlled by surface tension, in solid phase
+transitions the dominance of elastic long-range interactions leaves to surface tension only a
+minor role of a scale selection.
+First order phase transitions in liquids are well understood at both physical and mathe-
+matical level [32, 9]. The reason is that the scalar problem confronted in the liquid case is
+fully solvable [7]. Instead, despite many dedicated efforts, largely inspired by the pioneering
+contributions of J. Ericksen himself [12, 11, 13, 14, 15], the mathematical understanding of
+elastic phase transitions in solids is still far from being complete as the underlying nonconvex
+vectorial problems of the calculus of variations remain highly challenging.
+To set the stage, we recall that in nonlinear elasticity the energy functional can be written
+in the form E[y] =
+�
+Ω W(F )dx, where F = ∇y and y : Ω → Rn is the deformation. For
+the energy minimizing configurations the conventional physically informed energy density
+W(F ) can be replaced by a relaxed one QW(F ) = infφ∈C∞
+0 (D;Rn) |D|−1 �
+D W(F + ∇φ)dx
+which is known as quasiconvexification of W(F ) [8]. To construct the function QW(F )
+one must know the energy minimizing phase microstructures.
+In the case of liquids the
+geometry of such microstructures is irrelevant and the construction of QW(F ) reduces to
+convexification. In solids the task of finding the equilibrium microstructures in a generic
+case is hardly tractable [4, 1].
+With the aim of building a bridge between elastic phase transitions in liquids and solids,
+we consider a special limit of ‘near-liquid’ solids which are characterized by an arbitrarily
+weak resistance to shear. While we pose the general question of how in such a limit the tight
+control on the geometry of optimal microstructures by elastic interactions is progressively
+lost, we address a simpler problem of describing in this limit the boundary of the set of
+stable single-phase configurations. In the case of liquid-liquid phase transitions the incipient
+microstructures do not have any special features. The problem also simplifies in the case
+of ‘strongly-solid’ elastic phase transitions when the equilibrium microstructures are just
+simple laminates [24]. The goal of the present paper is to understand the opposite, ‘weakly-
+solid’ limit, when some of the simplest laminate-based microstructures are proved to be
+suboptimal.
+In the physics of phase transformations, the Maxwell-Gibbs critical/equilibrium condi-
+tions [34, 16], defining the incipient transitions in liquids, are designed to account for the
+possibility of phase nucleation. In other words, their role is to delimit the homogeneous
+configurations that are unstable to perturbations that are small only in extent and the set of
+such configurations is known in physics as the binodal region [36]. From the perspective of
+the mathematical theory of elastic phase transitions the analog of the binodal region would
+incorporate the homogeneous states that fail to be strong minima of the energy functional.
+Therefore, the binodal region is a subset in the configurational space of strain measures
+where the quasi convex envelope lays below the energy density. Locating the boundaries
+of the binodal region (known simply as a binodal) in the ’near-liquid’ limit constitutes the
+main task of the present paper. While remaining nontrivial, this task appears, a priori as
+more tractable than the task of constructing the actual quasiconvex envelope.
+In our prior work we have developed a general method for identifying the subsets of the
+2
+
+d1
+d2
+d
+h(d)
+Figure 1: Double-well structure of the energy density h.
+binodal supporting the laminate type energy minimizing configurations [18, 23, 25]. Behind
+this method is the study of stability of the jump set—a codimension one variety in the phase
+space that has a dual nature. On the one hand it determines the set of pairs F± that could
+be the traces of the deformation gradient at the phase boundary in a stable configuration.
+On the other, the jump set consists of points that are at most marginally stable in the sense
+that their every neighborhood contains points where quasiconvexity fails. Therefore, if one
+can prove quasiconvexity at a point on the jump set, then this point must lie on the binodal.
+In addition, we have also developed tools to constrain the location of the binodal by means
+of addressing nucleation phenomenon directly [19]. As we show in this paper, combined
+together, these two types of approaches can produce in the ’near-liquid’ limit a rather good
+practical understanding of the whole structure of the binodal, and even allow one to obtain
+the exact formulas for the quasiconvex envelope.
+To highlight ideas we focus here only on the simplest family of non-quasiconvex energy
+densities known as Hadamard materials [27, 28]: W(F ) = µ
+2|F |2 + h(det F ). Specifically,
+we’ll be interested in the case of two space dimensions and assume that the function h(d)
+describes a generic double-well potential modeling isotropic-to-isotropic phase transitions
+(see Fig. 1). The main advantage of this class of elastic materials is that one can identify
+a single parameter µ, scaling the effective rigidity; by varying this parameter we can study
+the entire range of intermediate rigidity responses from ’strong’ (µ ≫ 1) to ’weak’ (µ ≪ 1).
+A notable feature of the Hadamard materials is that the phase with the larger value of det F
+(smaller density) is characterized by a larger tangential (effective) rigidity than the phase
+with the smaller value of det F (larger density). As a result, the latter is more ’liquid-like’
+than the former and therefore the incipient phase transformation induced by compression can
+be expected to be different from the incipient phase transformation induced by stretching.
+As we show in what follows, this asymmetry leads to a coexistence of ’strongly-solid’ and
+’weakly-solid’ responses inside a single material model as, even in the absence of hysteresis,
+the direct and reverse solid-solid phase transitions proceed according to morphologically
+different transformation mechanisms.
+While for an Hadamard material the double well energy structure is described by the
+simplest scalar potential h(d), the results of relaxation of W(F ) are nontrivial due to the
+inherent incompatibility of the energy wells [3].
+We recall that W(F ) is quasiconvex if
+and only if h(d) is convex [2]. The relaxation of W(F ) with non-convex h(d) is known for
+3
+
+the ‘infinitely-weak’ solids (effectively fluids) with µ = 0, where QW(F ) = h∗∗(det F ) [7].
+Previously we explicitly constructed the quasiconvex envelope for W(F ) in the ‘strongly-
+solid’ limit assuming that the shear modulus µ is sufficiently large and the corresponding
+quadratic term dominates the double-well term. In this case the formula for QW(F ) couples
+|F | and det F and the relaxed energy is sandwiched between W(F ) above and U(F ) =
+µ
+2|F |2 + h∗∗(det F ) [24].
+In this paper we show that the constraint on µ in [24] was not a technical limitation,
+and that, as µ decreases, our formula for QW(F ) ceases to be valid in the subsets of the
+binodal region close to the ’liquid-like’ phase with smaller rigidity. In the limit of small µ,
+we show that the relaxation of W(F ) goes through a chain of structural transitions with
+simple lamination persisting only in the vicinity of the pure ’solid-like’ phase, being replaced
+by very different phase arrangements close to the ’liquid-like’ phase.
+Our main technical approach is to generate bounds on the binodal surface.
+The simplest bounds is obtained by probing the binodal by means of nucleating first
+rank laminates. Their optimality is proved by establishing their polyconvexity (and therefore
+quasi-convexity). In this setting this is an algebraic problem, because the supporting null-
+Lagrangians can be constructed explicitly, [25]. In contrast with the strongly solid regime of
+large µ analyzed in [24], in the near liquid regime of small µ, not all of the first rank laminate
+bounds are optimal.
+These bounds are then improved by nucleating second rank laminates. However, as shown
+in [26], the second rank laminate bounds are not optimal either, and are further improved
+for hydrostatic strains by means of nucleating a bounded circular inclusion in the infinite
+plane. We conjecture that this bound is optimal. If our conjecture is true, then the values
+of the deformation gradient in the exterior of the circular nucleus would provide a bound
+on the binodal from the outside of the binodal region. Another consequence of the assumed
+optimality of the inclusion-based nucleation bound is the explicit formula for the quasiconvex
+envelope QW(F ) at all hydrostatic strains.
+By juxtaposing the hypothetical bound provided by the study of bounded inclusions and
+unbounded second rank laminates we derive tight two-sided bounds on the binodal. As we
+demonstrate in [20], both bounds remain tight in the full range of parameters for which the
+bounds are meaningful. Moreover, the hypothetical bound being in complete agreement with
+the numerically computed rank-one convex binodal.
+The paper is organized as follows. In Section 2 we recall some general results from the
+calculus of variations for nonconvex vectorial problems, used in the rest of the paper. In
+Section 3 we specialize these results for the Hadamard material and present the numerical
+illustrations of the obtained bounds. Analytical results for the limiting case µ → 0 are pre-
+sented in Section 4 where we also compare them with numerical computations. In Section 5
+we demonstrate the far reaching consequences of the assumed optimality of the nucleation
+bound. The paper ends with a general discussion and conclusions in Section 6.
+4
+
+2
+Preliminaries
+Binodal region. Hyperelastic materials in a d-dimensional space have the following form of
+the energy stored in the deformed elastic body
+E[y] =
+�
+Ω
+W(∇y(x))dx,
+where Ω ⊂ Rd is the reference configuration, and y : Ω → Rd is the deformation.
+In
+order to understand the stable (i.e. experimentally observable) configurations of the body
+it is often necessary to replace the energy density W(F ) with a relaxed one QW(F ), called
+quasiconvexification. Even though, there is a formula for QW(F ) [8]:
+QW(F ) =
+inf
+φ∈C∞
+0 (D;Rn)
+1
+|D|
+�
+D
+W(F + ∇φ)dx,
+(2.1)
+there is no systematic approaches to compute it. A simpler, but just as useful an object, is
+the elastic binodal.
+Definition 2.1. An elastic binodal is the boundary of the binodal region
+B = {F : W(F ) < QW(F )}.
+(2.2)
+Definition 2.2. We say that the matrix F is stable, if W(F ) = QW(F ).
+Thus, the binodal is the boundary separating the binodal region from the set of stable
+points.
+Jump set. While we acknowledge that there could be rank-one convex, non quasiconvex
+functions, most cases of practical interest in elastic phase transitions feature multiwell ener-
+gies that are not rank-one convex and possess a non-trivial jump set, stable points of which
+form a part of the binodal (or the entire binodal, if one is lucky). The jump set is the set of
+solutions F = F− of the equations
+
+
+
+
+
+
+
+
+
+F+ = F− + a ⊗ n,
+[[P ]]n = 0,
+[[P T ]]a = 0,
+[[W]] − ⟨{{P }}, [[F ]]⟩ = 0,
+(2.3)
+where a ̸= 0 and |n| = 1 are thought to be excluded from the above system resulting in a
+single scalar equation for F . We refer the reader to [26] for a discussion of the geometry of
+the solution set of (2.3). Here we used the standard notations
+P± = WF (F±),
+[[F ]] = F+ − F−,
+{{P }} = P+ + P−
+2
+,
+⟨A, B⟩ = Tr (ABT),
+where WF indicates the matrix of partial derivatives Pij = ∂W/∂Fij.
+5
+
+The points on the jump set belong either to the binodal or to the binodal region B, [18].
+Hence, the jump set always represents a bound on the binodal region from within. One of
+the easy ways to detect the unstable parts of the jump set is to use the Weierstrass condition,
+which is necessary for stability.
+W ◦(F , b ⊗ m) ≥ 0,
+∀b ∈ Rn, |m| = 1,
+(2.4)
+where
+W ◦(F , H) = W(F + H) − W(F ) − ⟨WF (F ), H⟩.
+We have proved in [22] that the pairs of points F± on the jump set are either both stable
+or both unstable. Hence, a point F+ satisfying (2.4) can be still classified as unstable, if F−
+fails (2.4). While there are other conditions of stability that don’t follow from (2.4) (see [23])
+we will only make use of an easily verifiable corollary of(2.4) that restricts the rank-one test
+fields b ⊗ m to an infinitesimally small neighborhood of [[F ]] = a ⊗ n.
+Currently, the only general tool for establishing stability is polyconvexity, which is suf-
+ficient but rather far from necessary. In two dimensions it reduces to finding a constant
+m ∈ R, such that
+W ◦(F , H) − m det H ≥ 0,
+∀H ∈ R2×2.
+(2.5)
+If (2.5) holds, then F is stable in the sense of Definition 2.2. For points F± on the jump set,
+however, the only value of m that could possibly work is, as shown in [25],
+m = ⟨[[P ]], cof[[F ]]⟩
+|[[F ]]|2
+.
+(2.6)
+Secondary jump set. An improved bound on the binodal is provided by the secondary
+jump set corresponding to the nucleation of a rank-two laminate in the infinite homoge-
+neously strained space. Thus, the secondary jump set is defined by the system of equations
+
+
+
+
+
+
+
+
+
+
+
+F = F + b ⊗ m,
+P m = P m,
+P Tb = P
+Tb,
+W(F ) − W = P m · b,
+(2.7)
+where the pair F±, to be determined, is assumed to satisfy the primary jump set equations
+(2.3), while
+W = λW(F+) + (1 − λ)W(F−),
+P = λP+ + (1 − λ)P−,
+(2.8)
+for some λ ∈ [0, 1], which also plays the role of a variable to be solved for in (2.7), along
+with F , b ̸= 0, and |m| = 1. Once again, the secondary jump set represents a bound on the
+binodal region from within.
+Nucleation criterion. Let us now recall another method of probing the binodal: nucleation
+of inclusions either of a prescribed shape [5, 33, 31] or of an optimal inclusion, whose shape
+must be determined [29, 35, 30]. The theory justifying why these tests probe the binodal
+6
+
+was developed in [19]. In the case of “nucleation of a bounded inclusion”, the criterion for
+F0 to be “marginally stable”, i.e. to lie in the closure of B, is the existence of a field
+φ ∈ S = {φ ∈ L2
+loc(Rd) : ∇φ ∈ L2(Rd; Rd)},
+such that
+∇ · P (F0 + ∇φ) = 0,
+∇ · P ∗(F0 + ∇φ) = 0
+(2.9)
+in the sense of distribution in Rd, where
+P (F ) = WF (F ),
+P ∗(F ) = W(F )Id − F TP (F ).
+We also need to verify the non-degeneracy of the solution φ:
+�
+Rd W ◦
+F (F0, ∇φ)dx ̸= 0.
+(2.10)
+In the case of nucleation of an actual inclusion ω with smooth boundary the verification of
+(2.9) consists in verifying that the field φ ∈ S solves ∇ · P (F0 + ∇φ) = 0 both inside and
+outside of ω, together with the condition that the traces F±(x) = F0 + ∇φ±(x) on the two
+sides of ∂ω form a corresponding pair on the jump set for each x ∈ ∂ω. If, in addition, we
+can somehow prove that F + ∇φ(x) is stable in the sense of Definition 2.2, for each x ∈ Rd,
+then F0 must lie on the binodal. Conversely, if it is known that that at some x0 ∈ Rd the
+matrix F0 + ∇φ(x0) is unstable, then F0 must lie in the interior of B.
+3
+Hadamard material
+In this paper we focus our attention on a particularly simple, yet nontrivial energy
+W(F ) = µ
+2|F |2 + h(d),
+F ∈ {F ∈ GL(n) : det F > 0},
+d = det F ,
+(3.1)
+where h(d) is a C2(0, +∞) function with a double-well shape. In our explicit computations
+and illustrations we use the quartic double-well energy1
+h(d) = (d − d1)2(d − d2)2,
+(3.2)
+which affords certain simplification of general formulas.
+Jump set. We recall (see [24]) that in two dimensions the jump set of (3.1) consists of
+matrices F±, whose two singular values labelled ε0 and ε± satisfy the equations
+ε0[[h′]] + µ[[ε]] = 0,
+[[h]] − {{h′}}[[d]] = 0,
+d± = det F± = ε0ε±.
+(3.3)
+The notation reflects that for each pair F± on the jump set there is a frame in which
+both matrices are diagonal and share the same singular value ε0 with the same eigenvector.
+1Formula (3.2) only needs to hold in an arbitrary neighborhoodof [d1, d2]. The potential h(d) can be
+modified outside of that neighborhood arbitrarily, as long as h∗∗(d) = h(d) there. In particular, the singular
+behavior of h(d) as d → 0+, required in nonlinear elasticity, can be easily assured.
+7
+
+Equations (3.3) can be used to derive the semi-explicit parametric equations of the jump set,
+where, say d+ = ε0ε+, can serve as a parameter. Given d+ we can use the second equation in
+(3.3) to compute d− = D(d+). Then, multiplying the first equation in (3.3) by ε0 we obtain
+the parametric equations
+
+
+
+ε0(d+) =
+�
+−µ[[d]]
+[[h′]] ,
+ε+(d+) =
+d+
+ε0(d+).
+In the case of potential (3.2) we obtain
+[[h]] − {{h′}}[[d]] = [[d]]3(d1 + d2 − d+ − d−).
+Hence, d− = d1 + d2 − d+ = D(d+). It follows that
+{{h′}} = 0,
+ε+ + ε− = d1 + d2
+ε0
+.
+(3.4)
+In particular, we can eliminate h′(d±) from our formulas by means of (3.3) and (3.4):
+h′(d±) = {{h′}} ± 1
+2[[h′]] = ∓µ
+2
+[[ε]]
+ε0
+.
+(3.5)
+For quartic energy (3.2) we can also write the equation of the jump set explicitly as
+ε± = ε±(ε0). Indeed, ε± = d±/ε0, while d± solves
+(d± − d1)(d± − d2) = − µ
+4ε2
+0
+(3.6)
+The two roots of (3.6) are the values of d±, where, by convention, we denote by d+ the
+larger root. Equation (3.6) has exactly two real roots whenever ε0 > √µ/(d2 − d1). Hence,
+explicitly,
+ε± =
+1
+2ε0
+�
+d1 + d2 ±
+�
+(d2 − d1)2 − µ
+ε2
+0
+�
+.
+(3.7)
+In our calculations we will use equations (3.5) to eliminate all occurrences of h′(d±) and
+equations (3.7) to eliminate ε±, since the pair ε± is uniquely determined by a single parameter
+ε0.
+Numerical illustrations. When µ is large we have shown in [24] that the jump set [18]
+comprises the entire binodal, each point of which corresponds to the nucleation of a simple
+laminate, leading to an explicit formula for the relaxation QW(F ). As the shear modulus
+µ decreases, parts of the jump set will become unstable. The jump set will then undergo
+a topological change at µ = µtop and in the limit µ → 0, which is the main focus of this
+paper, a specific portion of it will remain stable, as we will show using methods from [25].
+Fig. 2 shows the jump sets and indicates their unstable parts for four different values of the
+shear modulus µ. The values of µ in Fig. 2 are chosen to be µ = 0, µtop/3, 0.9µtop, and
+1.5µtop. Dotted lines indicate “convexification hyperbolas”, i.e., hyperbolas ε2 = d1/ε1 and
+ε2 = d2/ε1, where the interval [d1, d2] is the interval on which h(d) differs from its convex hull.
+8
+
+1
+2
+3
+1
+0.5
+1
+1.5
+2
+2.5
+3
+2
+ = 0
+0.5
+1
+1.5
+2
+2.5
+1
+0.5
+1
+1.5
+2
+2.5
+2
+ = 2.8168
+0.5
+1
+1.5
+2
+1
+0.5
+1
+1.5
+2
+2
+ = 7.6617
+ = 12.6757
+1
+2
+3
+1
+0.5
+1
+1.5
+2
+2.5
+3
+2
+Figure 2: Jump sets for h(d) given by (3.2) with d1 = 1, d2 = 3, and different values of µ.
+All points outside of the region bounded by the convexification hyperbolas are well-known
+to be stable (see e.g. [9]), since they are obviously polyconvex.
+W-points. In [23] we have shown that the easily computable corollary of the Weierstrass
+condition (2.4) for the energy (3.1) has the form
+ε0 ≥ ε±.
+(3.8)
+In [24] we have shown that this condition is always satisfied for large values of µ as is evident
+from the lower right panel in Fig. 2, while it has an obvious geometric interpretation in the
+two panels in which the part of the jump set failing (3.8) is shown as a dashed line. The
+points marked by red dots in Fig. 2 that delimit the part of the jump set satisfying (3.8)
+will be called the Weierstrass points or W-points, for short. We have shown in [26] that the
+solid portion of the jump set delimited by W-points is polyconvex for all sufficiently small
+µ. As we show below, one can provide an almost explicit characterization of all values of µ
+for which W-points are also points of polyconvexity assuming the quartic nonlinearity (3.2).
+As discussed above, in order to prove the polyconvexity of W-points we need to establish
+(2.5), where m is given by (2.6). This problem has been already analyzed in [24], where we
+showed that (2.5) can be written as Φ(x, y) ≥ Φ(ε±, ε0) for all x, y, where
+Φ(x, y) = µ
+2(x2 + y2) − αx − βy − h(xy) − mxy,
+9
+
+α = 2√µR{{d}},
+β = µ2 + R4d+d−
+Rõ
+,
+m = [[h′d]]
+[[d]] ,
+R =
+�
+−[[h′]]
+[[d]] .
+According to the equations of the jump set (3.3) R = √µ/ε0. Hence, we also have
+α = µ(ε+ + ε−),
+β = µ
+�
+ε0 + ε+ε−
+ε0
+�
+,
+m = {{h′}} − µ{{ε}}
+ε0
+.
+When we minimized Φ(x, y) over all (x, y), such that xy = d we have concluded that the
+minimizer is (d/y, y), where y = y(d) is the largest root of
+y4 − β0y3 + dα0y − d2 = 0,
+α0 = ε+ + ε−,
+β0 = ε0 + ε+ε−
+ε0
+,
+(3.9)
+while the minimum of Φ(x, y) is achieved at a finite point corresponding to a critical point
+of φ(d) = Φ(d/y(d), y(d)).
+In the special case of W-points we have ε+ = ε0 and therefore α0 = β0 = ε− + ε0. In this
+case equation (3.9) factors
+(y2 − d)(y2 − α0y + d) = 0.
+The largest root is y = 1
+2(α0 +
+�
+α2
+0 − 4d), provided 0 < d ≤ α2
+0/4. If d > α2
+0/4, then the
+quartic has only two real roots y = ±
+√
+d. Thus,
+y(d) =
+�
+(α0 +
+�
+α2
+0 − 4d)/2,
+d ≤ α2
+0/4,
+√
+d,
+d > α2
+0/4.
+In [24] we have also computed
+φ′(d) = µy(d)2 − β0y(d)
+d
++ h′(d) − m.
+In the case of W-points for which β0 = α0 we see that
+y(d)2 − β0y(d)
+d
+= −1
+when d ≤ α2
+0/4. Hence, any critical points of φ(d) in this regime would have to satisfy
+h′(d) − µ − m = 0.
+One of the solutions is d−, which always satisfies d− ≤ α2
+0/4. If this equation has 3 solutions,
+the the middle one corresponds to a local maximum of φ(d), while the third d∗ > d+ always
+fails to satisfy d∗ ≤ α2
+0/4 because d+ = ε2
+0 > (ε− + ε0)2/4. We conclude that the only critical
+points of φ(d) that need to be checked are the ones that satisfy d > α2
+0/4, while
+φ′(d) = µ
+�
+1 − α0
+√
+d
+�
++ h′(d) − m.
+10
+
+Observe that φ′(d) > 0 when d ≥ max(α2
+0, �d+), where �d+ is the largest root of h′(d) − m.
+Hence we only need to check for critical points in a specific bounded interval. In fact, if h(d)
+is given by (3.2), then it is easy to see that φ′(d) > 0 for all d ≥ α2
+0. Hence, we only need to
+check for critical points of φ(d) on (α2
+0/4, α2
+0). In addition, since {{h′}} = 0 for h(d), given by
+(3.2) we have m = −µ{{ε}}/ε0 = −µα0/(2ε0). Thus, we obtain the following characterization
+of polyconvexity of W-points.
+Theorem 3.1. Let h(d) be given by (3.2), then W-points are polyconvex whenever
+min
+d∈
+�
+α2
+0
+4 ,α2
+0
+�
+�
+h(d) + µ
+�
+d + α0d
+2ε0
+− 2α0
+√
+d
+��
+= h(ε2
+0) − µε0
+�ε0
+2 + 3ε−
+2
+�
+.
+(3.10)
+where α0 = ε0 + ε−, with (ε0, ε−), (ε−, ε0), and (ε0, ε0) being the coordinates of W-points.
+The right-hand side in (3.10) is just φ(ε2
+0), where φ(d) is the function being minimized
+in (3.10). For quartic energy (3.2) we compute the coordinates of W-points by solving
+−4d(d − d1)(d − d2) = µ.
+Then ε2
+0 is the largest root, and
+ε− = d1 + d2 − ε2
+0
+ε0
+.
+We can compute the largest value of µ for which (3.10) holds by substituting µ = −4ε2
+0(ε2
+0 −
+d1)(ε2
+0−d2) into (3.10) and regarding ε0 ≤ √d2 as a parameter. When ε0 = √d2, φ(d)−φ(d2)
+is a positive polynomial in x =
+√
+d. We then seek numerically the largest value of ε0 <
+√d2 for which the polynomial P(x) = (φ(x2) − φ(ε2
+0))/(x − ε0)2 develops a double root.
+Algebraically this means seeking the largest root ε0 < √d2 of the discriminant (computed in
+Maple). This solution gives the largest value of µ below which the W-points are polyconvex.
+For example, when d1 = 1, d2 = 3, we have polyconvexity of W-points for all µ < 6.35888.
+In this paper we will be interested exclusively in the case when W-points are quasiconvex.
+In Fig. 2 the W-points are polyconvex in the top right panel and unstable in the bottom left
+panel.
+Secondary jump set. The algebraic equations (2.7) describing the secondary jump set can
+generally be solved only numerically. By contrast, when µ is small, the asymptotics of the
+solutions can be computed explicitly, providing an excellent approximation to the computed
+secondary jump set for µ < 3, with d1 = 1, d2 = 3. While the entire secondary jump set is
+unstable [26], we will see that it provides an excellent (inside) bound for the binodal.
+Suppose that F0 lies on the secondary jump set. Then there exists ε±, y and λ ∈ [0, 1],
+such that the pair F0, F , where
+F =
+�
+ε
+0
+0
+ε0
+�
+,
+ε = λε+ + (1 − λ)ε−,
+satisfies the jump set equations (2.7). We compute
+P = λP+ + (1 − λ)P− =
+�
+µε + h′ε0
+0
+0
+µε0 + εh′
+�
+.
+11
+
+We have
+P0 = µF0 + h′(d0)cofF0 = µ
+�
+ε
+0
+0
+ε0
+�
++ µb ⊗ m + h′(d0)
+��
+ε0
+0
+0
+ε
+�
++ b⊥ ⊗ m⊥
+�
+.
+Thus, the second and the third equations in the jump set system (2.7) become
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+ (h′(d0) − h′)ε0
+0
+0
+h′(d0)ε − εh′
+
+ m = −µb,
+
+ (h′(d0) − h′)ε0
+0
+0
+h′(d0)ε − εh′
+
+ b = −µ|b|2m.
+These equations result in 3 possibilities
+(a) (h′(d0) − h′)ε0 = h′(d0)ε − εh′ = −γ, µb = γm, m ∈ S1
+(b) (h′(d0) − h′)ε0 = −(h′(d0)ε − εh′) = −γ, µb = γI−m, I− =
+�
+1
+0
+0
+−1
+�
+, m ∈ S1
+(c) (h′(d0) − h′)ε0 ̸= ±(h′(d0)ε − εh′)
+Possibility (c) implies that F0 must be diagonal, and will be our main focus. In [26] we show
+that possibilities (a) and (b) have no solutions. Let us therefore assume that F± is diagonal
+and has the form
+F± =
+�
+ε±
+0
+0
+ε0
+�
+.
+This implies that F − F0 = βe2 ⊗ e2. In particular
+F0 =
+�
+x0
+0
+0
+y0
+�
+,
+x0 = ε = λε+ + (1 − λ)ε−,
+λ ∈ (0, 1).
+Let us compute the diagonal matrices P± using equations (3.4) and (3.5).
+P 11
+± = µε± + h′(d±)ε0 = µ{{ε}} = µ(d1 + d2)
+2ε0
+,
+P 22
+± = µε0 + h′(d±)ε± = µ
+�
+ε0 ∓ [[ε]]ε±
+2ε0
+�
+.
+Let us compute the diagonal matrix P0.
+P 11
+0
+= µx0 + h′(d0)y0 = µε + h′(d0)d0
+ε ,
+P 22
+0
+= µy0 + h′(d0)x0 = µd0
+ε
++ h′(d0)ε.
+Traction continuity equation (P − P0)e2 = 0 then becomes
+ε0 + [[ε]]
+2ε0
+(ε− − 2λ{{ε}}) − d0
+ε − h′(d0)
+µ
+ε = 0.
+12
+
+It will be convenient to use ε as a variable in place of λ. Replacing λ above using ε = ε−+λ[[ε]]
+we obtain
+d0
+ε = ε0 + 1
+ε0
+(ε+ε− − {{ε}}ε) − h′(d0)
+µ
+ε.
+(3.11)
+Let us now compute all the terms in the last equation in (2.7).
+W(F0) = µ
+2 (ε2 + y2
+0) + h(d0) = µ
+2
+�
+ε2 + d2
+0
+ε2
+�
++ h(d0).
+Next we compute
+W = W− + λ[[W]] = W− + λµ[[ε]]{{ε}} = W− + µ(ε − ε−){{ε}},
+where [[h]] = −{{h′}}[[d]] = 0 has been used. We compute
+h(d−) = [(d− − d1)(d− ��� d1)]2 =
+µ2
+16ε4
+0
+,
+according to (3.6). Therefore,
+W− = µ
+2 (ε2
+− + ε2
+0) + µ2
+16ε4
+0
+.
+We then compute F0 − F = (y0 − ε0)e2 ⊗ e2. Therefore
+⟨P , F0 − F ⟩ = µ
+�d0
+ε − ε0
+� �
+ε0 + 1
+ε0
+(ε+ε− − {{ε}}ε)
+�
+.
+Finally, the Maxwell equation W(F0) − W = ⟨P , F0 − F ⟩ can be written as
+1
+2
+�
+ε2 + d2
+0
+ε2
+�
++h(d0)
+µ
+−(ε−ε−){{ε}}−1
+2(ε2
+−+ε2
+0)− µ
+16ε4
+0
+=
+�d0
+ε − ε0
+� �
+ε0 + 1
+ε0
+(ε+ε− − {{ε}}ε)
+�
+.
+(3.12)
+Next we replace in the above form of the Maxwell relation the expression d0/ε by its expres-
+sion from (3.11) As a result of such a substitution the Maxwell relation will also become a
+quadratic equation in ε. This permits us to eliminate this variable as a rational expression in
+terms of ε0 and d0, while the Maxwell relation will also reduce to a rational relation between
+d0 and ε0. This calculation can only be done with the aid of a computer algebra system,
+since the remaining equation F(ε0, d0) = 0 is very long and complicated. Now for a given
+choice of numerical values of µ, d1 and d2 we can solve F(ε0, d0) = 0 numerically and then
+extract those solutions which satisfy λ ∈ [0, 1]. The result for d1 = 1, d2 = 2 and µ = µtop/3
+is shown as a green curve in Fig. 3. As we can see, it identifies all points between the green
+curve and the dashed lines of the primary jump set as a part of the binodal region—an
+improvement over the primary jump set bound.
+While, it is not apparent from Fig. 3, the secondary jump set consists of two curves
+related by symmetry with respect to the bisector of the first quadrant. Each of the curves
+13
+
+0.5
+1
+1.5
+2
+2.5
+1
+0.5
+1
+1.5
+2
+2.5
+2
+ = 2.8168
+Figure 3: Secondary jump set computed by numerically solving equations (3.11) and (3.12).
+are cut-off at their intersection with each other at the bisector. Each curve starts at a W-
+point and ends at a point (not shown) on the dashed part of the jump set. The endpoints
+of the secondary jump set correspond to the extreme values 0 and 1 of the volume fraction
+λ in (2.8). The corresponding points on the secondary jump set must lie on the primary
+jump set. There are two possibilities. Either F ̸= F or F = F at λ = 0 or 1. In the
+former case the limiting position F+ of F is rank-one related to two different points on the
+jump set: F− (layer normal e1) and F (layer normal e2). W-point F+ is the only one with
+this property. All other points F+ on the jump set have a unique counterpart F−. In the
+latter case a detailed asymptotic analysis shows that that the common limit point of F and
+F must achieve equality in the “Legendre-Hadamard for phase boundaries” inequality from
+[23]. This point lies on the dashed part of the jump set and is used in numerical calculations.
+The details of the analysis will be spelled out in a forthcoming paper [20].
+Circular nucleus. In [26] it is shown that the secondary jump set (green curve in Fig. 3)
+is unstable. That means that the corresponding bound on the binodal is not optimal. We
+can improve the bound using another method of probing the binodal: nucleation of equilib-
+rium energy-neutral inclusions. The theory justifying why such nucleation tests probe the
+binodal was developed in [19]. In the case of the isotropic, objective energy (3.1) and a
+hydrostatic loading it is natural that the shape of an optimal precipitate should be circular.
+The deformation gradient inside the circular precipitate must be a constant hydrostatic field
+F0 = εW
+0 I2, since that field is rank-one connected to an infinite family of fields
+FR = R
+�
+εW
+−
+0
+0
+εW
+0
+�
+RT,
+R ∈ SO(2),
+where (εW
+− , εW
+0 ) is a coordinate of one of the W-points. The deformation gradient outside of
+the circular inclusion must solve an Euler-Lagrange equation for the energy (3.1)
+µ∆y + (cof∇y)∇h′(det ∇y) = 0,
+x ∈ R2 \ B(0, 1),
+(3.13)
+14
+
+and agree with FR at the point Re1 on the boundary of the circular inclusion:
+∇y(x) = εW
+− n ⊗ n + εW
+0 τ ⊗ τ,
+x ∈ ∂B(0, 1).
+(3.14)
+In this case both equations (2.9) will be satisfied for the possibly marginally stable matrix
+F∞ = lim
+|x|→∞ ∇y(x) = ε∞I2.
+We also know that that the values of ∇y(x) inside the circular inclusion and its trace on
+the outside boundary of the inclusion are stable. Our results from [19] then say that either
+F∞ lies on the binodal and all values ∇y(x) in the exterior of the inclusion are stable, or
+F∞ lies in the interior of the binodal region B.
+In our special radially symmetric case we look for a radially symmetric solution of (3.13)
+y = η(r)ˆx,
+|x| > 1.
+The the unknown function η(r) must solve
+�
+η
+r
+d
+drh′ �
+ηη′
+r
+�
++ µ
+�
+η′ + η
+r
+�′ = 0,
+r > 1,
+η′(1) = εW
+− ,
+η(1) = εW
+0 .
+(3.15)
+The nonlinear second order ODE (3.15) cannot be integrated explicitly, but can be solved
+numerically. In order to do so, we need to convert the infinite range r > 1 into a finite one
+by means of the change of the independent variable x = 1/r2. It will also be convenient to
+change the dependent variable v = η/r, so that v(x) would have a finite limit, when x → 0.
+Then v(x) solves
+v′′ = − (v′)2vh′′(v2 − 2xvv′)
+µ + v2h′′(v2 − 2xvv′),
+x ∈ [0, 1],
+v(1) = εW
+0 ,
+v′(1) = εW
+0 − εW
+−
+2
+.
+(3.16)
+The value ε∞ = v(0)I2 found numerically is shown as a blue dot in Fig. 4. It provides an
+improved bound on the binodal compared to the secondary jump set (green line in Fig. 4) by
+showing that hydrostatic strains between the blue dot and the green line are unstable. This
+conclusion holds, provided the non-degeneracy condition (2.10) is verified. A calculation
+shows that
+�
+Rd W ◦
+F (F0, ∇φ)dx = −I2
+�
+R2 h′′(ε2
+∞)ε2
+∞
+�
+η′(r) + η(r)
+r
+− 2ε∞
+�
+dx.
+Thus,
+�
+Rd W ◦
+F (F0, ∇φ)dx = −2πh′′(ε2
+∞)ε2
+∞I2 lim
+r→∞(rη(r) − ε∞r2).
+To see that the limit above exists and is non-zero, at least for small µ > 0, we simply solve
+(3.15) for µ = 0, for which εW
+0 = √d2, εW
+− =
+d1
+√d2. The solution is η(r) = √d1r2 + d2 − d1,
+and we easily see that
+lim
+r→∞(rη(r) − ε∞r2) = d2 − d1
+2√d1
+.
+15
+
+Hence, the non-degeneracy condition (2.10) will hold, at least for sufficiently small µ > 0.
+The non-degeneracy will also hold for all µ below the topological transition, because if we
+write �η(r) = η(r) − ε∞r, then (assuming that �η′(r) → 0, as r → ∞) �η(r) will solve, when r
+is large, the differential equation
+ε∞h′′(ε2
+∞)
+�
+ε∞
+�
+�η′ + �η
+r
+�
++ �η′�η
+r
+�
++ µ
+�
+�η′ + �η
+r
+�
+= 0.
+This integrates to
+ε∞h′′(ε2
+∞)(2ε∞r�η + �η2) + 2µr�η = 2c.
+Since �η, satisfying �η′(r) → 0, as r → ∞, cannot be zero (it is the leading term of η(r)−ε∞r),
+we conclude that the constant of integration c cannot be zero either. Hence, we obtain that
+lim
+r→∞(rη(r) − ε∞r2) = lim
+r→∞ r�η(r) =
+c
+µ + ε2∞h′′(ε2∞) ̸= 0.
+Polyconvexity limits along εI2. We now turn to the problem of proving polyconvexity at
+points F = εI2. To succeed we need to find a constant m ∈ R, such that (2.5) holds. For
+our energy we compute
+W ◦(F , H) = µ
+2|H|2 + h(ε2 + d + εθ) − h(ε2) − εh′(ε2)θ,
+θ = Tr H, d = det H.
+We also have
+|H|2 = 1
+2|H − HT|2 − 2d + θ2.
+Hence we need to find m ∈ R, such that
+µθ2
+2
++ h(ε2 + d + εθ) − h(ε2) − εh′(ε2)θ ≥ (m + µ)d,
+∀d ≤ θ2
+4 .
+(3.17)
+In particular the inequality must hold for d = θ2/4. In that case we must have
+m ≤ µ + 4 min
+θ∈R
+h(ε2 + θ2/4 + εθ) − h(ε2) − εh′(ε2)θ
+θ2
+= m∗.
+(3.18)
+The infimum of the smooth function
+F(d, θ) = µθ2
+2
++ h(ε2 + d + εθ) − h(ε2) − εh′(ε2)θ − (m + µ)d
+must be attained either at a critical point or at infinity. Obviously, if along the minimizing
+sequence the quantity δ = ε2 + d + εθ goes to infinity, then the values of F must also go to
++∞, which cannot happen along a minimizing sequence. Hence, (d, θ) go to infinity so that
+δ stays bounded. Hence, we can switch variables and instead of the pair (d, θ) consider the
+pair (δ, θ). In this case d = δ − εθ − ε2. Hence,
+F(δ, θ) = µθ2
+2
++ h(δ) − h(ε2) − εh′(ε2)θ − (m + µ)(δ − εθ − ε2),
+16
+
+where δ ≤ (θ/2 + ε)2. Minimizing F(δ, θ) with respect to θ we obtain
+θ = −ε(m + µ − h′(ε2))
+µ
+.
+Hence we need to minimize
+f(δ) = h(δ) − h(ε2) − (m + µ)(δ − ε2) − ε2(m + µ − h′(ε2))2
+2µ
+,
+over all δ satisfying
+δ ≤ ε2(h′(ε2) + µ − m)2
+4µ2
+.
+(3.19)
+We remark that taking θ = −4ε in (3.18) we conclude that m∗ ≤ µ + h′(ε2). Thus, the
+right-hand side of (3.19) is monotone decreasing in m, when m ≤ m∗.
+Now, the minimum is achieved either at the boundary, where equality in (3.19) holds, or
+at a critical point. It cannot be “achieved” at infinity, where f(δ) is +∞. If the minimum
+is achieved at the boundary then f(δ) ≥ 0 for all δ, provided m ≤ m∗. If the minimum is
+achieved at a critical point
+h′(δ) = m + µ,
+then several possibilities need to be considered. Let us first assume that the equation
+h′(δ) = m∗ + µ
+(3.20)
+has a single root δ∗. If that root fails to satisfy (3.19) with m = m∗, then for m = m∗ the
+function f(δ) has no critical points and polyconvexity holds. If that root satisfies (3.19),
+then all values δ < δ∗ are admissible. We then substitute m = h′(δ) − µ, δ ≤ δ∗ into f(δ).
+The resulting function
+f(δ) = h(δ) − h(ε2) − h′(δ)(δ − ε2) − ε2(h′(δ) − h′(ε2))2
+2µ
+(3.21)
+can be plotted on (−∞, δ∗] versus m = h′(δ) − µ to see if there are values of m above which
+all values of f(δ) are positive.
+Yet another possibility is when equation (3.20) has 3 real roots. If even the smallest root
+δ∗ fails to satisfy (3.19) with m = m∗, then there are no critical points and polyconvexity
+holds. Otherwise, all values of δ ≤ δ∗ are admissible and we can prove failure of polyconvexity
+by plotting (3.21) versus m(δ) = h′(δ)−µ on δ ≤ δ∗ and checking that it is negative. In fact,
+we believe that polyconvexity fails in all cases when δ∗ satisfies (3.19). In order to exhibit
+this failure we only need to produce a single value of admissible δ for which f(δ), given
+by (3.21), is negative. Hence, if (3.21) is negative for all δ ≤ δ∗, then there is no need to
+examine other intervals of admissible δ, since for any m ≤ m∗ there is always an admissible
+δ ≤ δ∗, which makes f(δ) negative. However, if f(δ) has a region where it is positive, then
+one needs to examine other areas of admissibility and check whether f(δ) is negative for the
+same values of m. Thus, we obtain an algorithm that can prove polyconvexity or failure of
+17
+
+1.1
+1.15
+1.2
+1
+1.06
+1.08
+1.1
+1.12
+1.14
+1.16
+1.18
+1.2
+2
+ = 2.8168
+nucleation bound
+polyconvexity bound
+secondary jump set
+Figure 4: Bounds on the binodal from the inside and the outside of the binodal region along
+hydrostatic strains.
+it in many, but not all cases. Polyconvexity holds whenever
+δ∗ > ε2(h′(ε2) + µ − m∗)2
+4µ2
+.
+for all solutions δ∗ of h′(δ) = µ + m∗. Polyconvexity fails whenever f(δ) < 0 for all δ < δ∗,
+provided
+δ∗ ≤ ε2(h′(ε2) + µ − m∗)2
+4µ2
+.
+If ε2 = d1, then the minimization problem (3.18) simplifies:
+min
+θ∈R
+h(d1 + θ√d1 + θ2/4)
+θ2
+.
+We first observe that in general θ = 0 is not a minimizer. Then there are 3 minimizers:
+θ = −4
+�
+d1,
+θ = ±2
+�
+d2 − 2
+�
+d1.
+When ε = √d1 + x, then the minimizer θ(x) must be located near one of the above 3
+minimizers. We can then write θ = θ0 + y for the minimizer, where θ0 denotes one of the 3.
+If we write the function under the minimum as H(ε, θ), then at the minimum we must have
+∂H/∂θ = 0, which gives the equation
+x ∂2H
+∂θ∂ε + y∂2H
+∂θ2 = 0.
+After solving for y and substituting this solution back into H we obtain
+H = x
+
+
+∂H
+∂ε − ∂H
+∂θ
+∂2H
+∂θ∂ε
+∂2H
+∂θ2
+
+
+ ,
+18
+
+where derivatives are evaluated at (√d1, θ0). Maple calculation yields
+H =
+
+
+
+
+
+x
+2
+√d1h′′(d1),
+θ0 = −4√d1,
+xd1h′′(d1)
+√d1+√d2 ,
+θ0 = −2√d2 − 2√d1,
+xd1h′′(d1)
+√d1−√d2 ,
+θ0 = 2√d2 − 2√d1.
+This shows that θ = 2√d2 − 2√d1 + y is the minimizer, while
+m∗ = µ −
+4xd1
+√d2 − √d1
+h′′(d1).
+In particular, the equation h′(δ) = m∗ + µ will have 3 real roots. The smallest one δ∗ will
+be near d1:
+δ∗ = d1 + µ + m∗
+h′′(d1) .
+Finally, polyconvexity will hold if (3.19) fails when δ = δ∗ and m = m∗. In other words we
+must have (asymptotically)
+ε ≤
+�
+d1 +
+µ
+h′′(d1)√d1
+√d2 − √d1
+√d2 + √d1
+.
+(3.22)
+Fig. 4 showing the right-hand side of (3.22) as a red dot implies that ε∞I2 fails to be
+polyconvex, but by a very slim margin. The ordering of the bounds in Fig. 4 persists on
+the entire range of µ. We see that the gap between established stability (along the bisector
+below the red dot) and established instability (along the bisector above the blue dot) is very
+small.
+4
+Limiting case µ → 0
+In this section we derive explicit asymptotics of the secondary jump set and the nucleation
+bound.
+Secondary jump set. Expanding equation (3.7) to first order in µ we obtain
+ε+ = d2
+ε0
+−
+µ
+4ε3
+0(d2 − d1) + O(µ2),
+ε− = d1
+ε0
++
+µ
+4ε3
+0(d2 − d1) + O(µ2).
+(4.1)
+When d1 and d2 are fixed we think of ε± as functions of ε0 and µ, even if we suppress this
+in the notation. Clearly, when µ → 0 we have ε+ → d2/ε0, ε− → d1/ε0.
+The parametric equations (x0(ε0; µ), y0(ε0; µ)) of secondary jump set converge, when
+µ → 0, to the hyperbola x0y0 = d1. In particular, d0(ε0, µ) → d1, as µ → 0. The volume
+fraction λ of the rank-one laminate used in the second rank laminate is also a function of
+ε0 and µ and must have a limit (at least along a subsequence) λ(ε0; µ) → λ0(ε0), as µ → 0.
+Equation (3.11) shows that d0 = d1 + µδ + O(µ2), while δ satisfies
+d
+ε0
+�
+ε0 + 1
+ε0
+�d1
+d2
+ε2
+0 − d1 + d2
+2ε2
+0
+d
+��
+− d1 − 2δ(d2 − d1)2d
+2
+ε2
+0
+= 0,
+(4.2)
+19
+
+where d = λd2 + (1 − λ)d1. Equation (4.2) was obtained simply by passing to the limit as
+µ → 0 in equation (3.11).
+When we pass to the limit as µ → 0 in (3.12) we obtain
+(d − d1)2(ε4
+0 + d
+2 − 2d2d)
+2ε2
+0d
+2
+= 0.
+(4.3)
+The dependence of d on the volume fraction λ is essential and should not disappear in the
+limit µ → 0. Therefore, the solution of (4.3) that we are after is
+d = d2 −
+�
+d2
+2 − ε4
+0,
+(4.4)
+where the choice of the root was dictated by the requirement that d ≤ d2. Combining this
+with the requirement that d ≥ d1 shows that
+4�
+d2
+2 − (d2 − d1)2 ≤ ε0 ≤
+�
+d2.
+(4.5)
+Substituting (4.4) into (4.2) gives the explicit formula for δ:
+δ = ε4
+0(d2 − d1) − 2(d2
+2 − ε4
+0)(d2 −
+�
+d2
+2 − ε4
+0)
+4ε2
+0(d2 − d1)2(d2 −
+�
+d2
+2 − ε4
+0)2
+.
+(4.6)
+It seem that in order to obtain the correct asymptotics of the secondary jump set we need
+to obtain the first order asymptotics of ε:
+ε = d2 −
+�
+d2
+2 − ε4
+0
+ε0
++ �εµ + O(µ2).
+(4.7)
+In fact, this is not necessary because the leading order asymptotics of d0 is a constant d1.
+In that case, as far as the first order asymptotics as µ → 0 is concerned, using (4.7) simply
+corresponds to reparametrizing the curve
+
+
+
+
+
+
+
+x0 = d2 −
+�
+d2
+2 − ε4
+0
+ε0
+,
+y0 = d1 + µδ(ε0)
+x0
+.
+(4.8)
+Indeed, if we change the curve parameter ε0 to ε0 + µ�ε/x′
+0(ε0), then
+x0
+�
+ε0 +
+µ�ε
+x′
+0(ε0)
+�
+= x0(ε0) + µ�ε + O(µ2).
+At the same time
+y0
+�
+ε0 +
+µ�ε
+x′
+0(ε0)
+�
+=
+d1
+x0(ε0) −
+µd1�ε
+x0(ε0)2 + µδ(ε0)
+x0(ε0) + O(µ2) = d1 + µδ
+x0 + µ�ε + O(µ2).
+20
+
+0
+1
+2
+3
+4
+5
+1
+1.05
+1.1
+1.15
+asymptotic
+numerical
+pcx
+asymptotic
+Figure 5: Comparison between the asymptotics (4.12) and ε∞ obtained from the numerical
+solution of (3.15).
+We conclude that equation (4.8) correctly describes the asymptotics of the secondary jump
+set with O(µ2) error, where the parameter ε0 varies according to (4.5). When ε0 = √d2,
+the secondary jump set enters one of the W-points, while when ε0 =
+4�
+d2
+2 − (d2 − d1)2 the
+secondary jump set enters its other end at the “Legendre-Hadamard for phase boundaries”
+bound that for small µ lies on the dashed part of the jump set in Fig. 3.
+The plot of
+(4.8) is in Fig. 3 and is indistinguishable from the numerically obtained curve using the full
+(non-asymptotic) versions of secondary jump set equations.
+Circular nucleus. In the near-liquid limit µ → 0 we can find the asymptotics of the
+solution explicitly.
+We know that in the limit µ → 0 the field d(x) = det ∇y(x) must
+approach d1. Hence,
+ηη′
+r
+= d1 + µδ(r) + O(µ2),
+r > 1.
+That implies
+η(r) =
+�
+d1r2 + c0 + µ�η(r) + O(µ2),
+(4.9)
+and therefore,
+δ(r) = 1
+r
+�
+�η(r)
+�
+d1r2 + c0
+�′
+.
+Substituting this ansatz into (3.15) we obtain
+µ
+√d1r2 + c0
+r
+h′′(d1)δ′(r) + µ
+�
+d1r
+√d1r2 + c0
++
+√d1r2 + c0
+r
+�′
++ O(µ2) = 0.
+(4.10)
+Initial conditions from (3.15) imply that
+c0 = d2 − d1,
+�η(1) = −
+d2 − d1
+4d3/2
+2 h′′(d2)
+,
+�η′(1) = d1(d2 − d1)
+2d3/2
+2
+�
+1
+d1h′′(d1) +
+1
+2d2h′′(d2)
+�
+.
+21
+
+Equation (4.10) is easy to integrate (observing that √d1r2 + c0/r is decreasing from √d2 to
+√d1 and is therefore uniformly bounded away from zero and ∞).
+h′′(d1)�η(r) =
+c1r2 + c2
+√d1r2 + c0
+−
+r2
+2√d1r2 + c0
+ln
+√d1r2 + c0
+r
+.
+(4.11)
+From initial conditions for �η(r) we obtain
+c1 = 1
+2 ln
+�
+d2,
+c2 = −(d2 − d1)h′′(d1)
+4d2h′′(d2)
+,
+and hence
+ε∞ =
+��
+d1 +
+µ
+2h′′(d1)√d1
+ln
+√d2
+√d1
+�
+I2 + O(µ2).
+(4.12)
+Figure 5 shows the quality of the asymptotics for the entire range of shear moduli µ. The
+numbers on the y-axis indicate that even for values of µ that are not particularly small the
+asymptotics (4.12) gives a good approximation of the actual value of ε∞. For example, for
+µ = 3 the relative discrepancy is only around 0.1%.
+5
+A glimpse into the relaxed energy
+Hypothetical bounds on the binodal. We have seen in the foregoing discussion that the energy
+W(F ) is not polyconvex at F = ε∞I2. This is not very surprising, since polyconvexity is
+usually strictly stronger that quasiconvexity and we expect and conjecture that F = ε∞I2
+lies on the binodal—at the very edge of quasiconvexity. Here we recall our observation that
+if someone could prove that F = ε∞I2 is stable, then we would immediately conclude that
+for every |x| > 1
+∇y(x) = η′(r)ˆx ⊗ ˆx + η(r)
+r (I2 − ˆx ⊗ ˆx)
+would be stable in the sense of Definition 2.2, providing a bound on the binodal from the
+outside. For the entire range of µ for which W-points are polyconvex the union of the curves
+�
+ε1 = η(r)
+r ,
+ε2 = η′(r),
+and
+�
+ε1 = η′(r)
+ε2 = η(r)
+r ,
+r > 1
+(5.1)
+are indistinguishable from the secondary jump set curves shown in green in Fig. 3. Fig. 6
+shows the same blown-up part of the strain space as in Fig. 4, where the curves (5.1) shown in
+magenta are passing through the blue point from Fig. 4. Assuming the conjectured stability
+of ε∞I2, the magenta curve must lie outside of binodal region, while secondary jump set
+lies in its interior [26]. Thus, the binodal of the energy (3.1) would have to lie between the
+green and the magenta curves. We will even go so far as to conjecture that the magenta
+curve is in fact the actual binodal of the energy (3.1). Regardless, under the assumption of
+stability of ε∞I2, the magenta line represents a rather tight outside bound on the binodal
+region. Another byproduct of the assumed stability of ε∞I2 would be the formula for the
+22
+
+0.5
+1
+1.5
+2
+2.5
+1
+0.5
+1
+1.5
+2
+2.5
+2
+ = 2.8168
+hypothetical binodal
+known binodal region
+1.1
+1.15
+1.2
+1
+1.06
+1.08
+1.1
+1.12
+1.14
+1.16
+1.18
+1.2
+2
+ = 2.8168
+nucleation bound
+polyconvexity bound
+secondary jump set
+outside bound
+Figure 6: A hypothetical bound on the binodal region from the outside, assuming stability
+of ε∞I2.
+quasiconvex envelope QW(F ) for hydrostatic strains F . If F = ε∞I2 is stable, then our
+radial solution ∇y(x) = η(r)ˆx of (3.15) is also a global minimizer in every finite ball B(0, R),
+where it satisfies the affine boundary condition y(x) = (η(R)/R)x, x ∈ ∂B(0, R) [21]. The
+energy of such configurations must necessarily be QW(η(R)I2/R)|B(0, R)|. This permits
+us to compute QW(εI2) for all ε, as the energy of y(x) = η(r)ˆx in B(0, R). Using the
+Clapeyron-type formula for the nonlinear elastic energy stored in an equilibrium stationary
+configuration we obtain for F = η(R)I2/R: [21]
+|B(0, R)|QW(F ) = 1
+2
+�
+∂B(0,R)
+{P (∇y)n · y + P ∗(∇y)n · x}dS.
+(5.2)
+Substituting n = ˆx, y = η(r)ˆx into (5.2) we obtain
+QW
+�η(R)
+R I2
+�
+= 2(µ − h′(d))d − µη′(R)2 + (2h′(d) + µ)η(R)2
+R2
++ 2h(d),
+(5.3)
+where
+d = η′(R)η(R)
+R
+.
+When µ is small we can use the explicit asymptotic formulas (4.9), (4.11) for η(r) to obtain
+an explicit asymptotics for QW(εI2). The plot of QW(εI2), coming from the numerical
+solution of (3.15), as well as its explicit asymptotic approximation, superposed on the plot
+of W(εI2) is shown in Fig. 7.
+6
+Conclusions
+In this paper our far reaching goal was to solve analytically the relaxation problem for the
+double well Hadamard energy (3.1) in two space dimensions when the rigidity measure µ
+23
+
+1
+1.2
+1.4
+1.6
+1.8
+3
+4
+5
+6
+7
+8
+9
+Energy
+W(  I2)
+QW(  I2)
+QWasym(  I2)
+Figure 7: Quasiconvex envelope of W(F ) restricted to hydrostatic strains F = εI2.
+is sufficiently small. An apparently more attainable target was to locate the corresponding
+binodal region inside the strain space. The study of the limit µ → 0 was expected to show how
+the ’cooperative’ , rigidity-controlled microstructures, dominating the quasiconvex envelope
+at large µ, give rise to more arbitrary and less controlled microstructures characterizing first
+order phase transitions in zero rigidity liquids.
+We used some of our previously developed methods to pinpoint a substantial portion
+of the binodal.
+While our general methods apply for Hadamard materials in the entire
+parameter range and are amenable to numerical implementation, here we were able to obtain
+the explicit asymptotic formulas only in the ’near-liquid’ regime. In particular, we showed
+that in an ’almost liquid’ limit, a subset of the jump set adjacent to the high strain phase
+remains stable which ensures that simple lamination delivers the corresponding part of the
+binodal. This means that even when the reference measure of rigidity µ is small, the high
+strain phase maintains its tangential rigidity at the level which ensures solid-solid like nature
+of the incipient phase transition. Instead, our analysis showed that the subset of the jump set
+adjacent to the low strain and low rigidity phase is unstable in the µ → 0 limit. Moreover, the
+secondary jump set is also unstable in this limit. This result suggests that laminates of any
+finite rank are unstable near the corresponding subset of the binodal. As we’ve demonstrated
+for hydrostatic strains, the reduced rigidity control in this range allows the incipient phase
+transformation to proceed non-cooperatively through the formation of isolated nuclei of the
+more rigid phase inside the matrix of the less rigid phase. Such transformation mechanism is
+already very similar to the one believed to be operating in purely fluid-fluid phase transitions.
+Whether the revealed asymmetry of the transformation mechanism between the direct
+and reverse transformation is a peculiarity of the Hadamard material or whether this striking
+phenomenon has a more general nature, remains to be established. It shows, however, the
+intricate role of rigidity in structural transformations which, even if weak, can produce rather
+complex structure of the relaxed energy. This complexity will then reflect a gradual transition
+from geometrically ordered microstructures, controlled by long range elastic interactions,
+24
+
+to more ’fluid’ microstructures whose spatial organization is mostly affected by molecular
+interactions operating at short range. In other words, in this limit the direct and reverse
+solid-solid phase transitions can operate through different transformation mechanisms. The
+fact that the ensuing complex structure of the relaxed energy at ’almost-liquid’ solid-solid
+phase transitions is ultimately replaced by a simple energy convexification at fluid-fluid phase
+transitions points to a singular nature of the limit µ → 0.
+Acknowledgments.
+YG was supported by the National Science Foundation under
+Grant No. DMS-2005538. The work of LT was supported by the French grant ANR-10-
+IDEX-0001-02 PSL.
+References
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+Neff, editors, Poly-, Quasi- and Rank-One Convexity in Applied Mechanics, pages 1–15.
+Springer Vienna, Vienna, 2010.
+[2] J. M. Ball and F. Murat. W 1,p-quasiconvexity and variational problems for multiple
+integrals. J. Funct. Anal., 58(3):225–253, 1984.
+[3] J.M. Ball and R.D. James. Incompatible sets of gradients and metastability. Archive
+for Rational Mechanics and Analysis, 218(3):1363–1416, 2015.
+[4] John M. Ball. Some open problems in elasticity. In Geometry, mechanics, and dynamics,
+pages 3–59. Springer, New York, 2002.
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+

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+Gravitational collapse of scalar and vector fields
+Karim Mosani,∗ Koushiki,† Pankaj S. Joshi,‡ and Jay Verma Trivedi§
+International Centre for Space and Cosmology, School of Arts and Sciences,
+Ahmedabad University, Ahmedabad-380009 (Guj), India.
+Tapobroto Bhanja¶
+International Center for Cosmology, & PDPIAS,
+Charotar University of Science and Technology, Anand- 388421 (Guj), India
+(Dated: January 13, 2023)
+We study here the unhindered gravitational collapse of spatially homogeneous (SH) scalar fields φ
+with a potential Vs(φ), as well as vector fields ˜A with a potential Vv(B) where B = g( ˜A, ˜A) and g is
+the metric tensor. We show that in both cases, classes of potentials exist that give rise to black holes
+or naked singularities depending on the choice of the potential. The strength of the naked singular-
+ity is examined, and they are seen to be strong, in the sense of Tipler, for a wide class of respective
+potentials. We match the collapsing scalar/vector field with a generalized Vaidya spacetime outside.
+We highlight that full generality is maintained within the domain of SH scalar or vector field collapse.
+keywords: Gravitational collapse, singularity, scalar field, vector field, causal structure.
+I.
+INTRODUCTION
+The contraction of a matter field under its gravita-
+tional influence is called gravitational collapse. In 1939,
+Oppenheimer and Snyder [1], and independently in 1938,
+Datt [2] developed the first solution of Einstein’s field
+equations (called the OSD model) depicting the gravita-
+tional collapse of a massive star. They considered a very
+specific case of spatially homogeneous (SH) dust collapse
+(By spatial homogeneity, we mean homogeneous on a
+three-dimensional spacelike orbit with a six-dimensional
+isometry group G6 corresponding to the spacetime [3]).
+Such a matter field undergoes gravitational collapse that
+ends up in a singularity. Such a spacetime singularity
+is hidden behind an event horizon, not visible to any ob-
+server, and what we obtain is a black hole as the outcome
+of continual collapse.
+Extending the above special scenario, in 1969, Penrose
+proposed what is now known as the cosmic censorship hy-
+pothesis (CCH) [4]. The weaker version of the hypothesis
+states that all singularities of gravitational collapse are
+hidden within a black hole and hence, cannot be seen
+by a distant observer (a globally naked singularity can-
+not exist). The strong version of the hypothesis states
+that no past inextendible nonspacelike geodesics can ex-
+ist between the singularity and any point in the space-
+time manifold. In other words, a causal geodesic with
+a positive tangent “at” the singularity does not exist (a
+locally naked singularity also cannot exist).
+The sup-
+porting argument for the validity of the strong CCH is
+the desirability of the spacetime manifold to be globally
+∗ kmosani2014@gmail.com
+† koushiki.malda@gmail.com
+‡ pankaj.joshi@ahduni.edu.in
+§ jay.verma2210@gmail.com
+¶ tapobroto.bhanja@gmail.com
+hyperbolic. Global hyperbolicity implies the existence of
+Cauchy surfaces embedded in the total manifold, thereby
+making general relativity a deterministic theory [5–7].
+Now singularity theorems of Hawking and Penrose
+[6, 11] do not imply that singularities are hidden from
+an external observer under any possible circumstances.
+In fact, singularity theorems take the causality condition
+as one of the axioms to start with to prove the existence
+of incomplete past (future) directed causal curves. Addi-
+tionally, the OSD model that motivated cosmic censor-
+ship is a special case. Joshi and Malafarina [12] showed
+that any arbitrarily small neighbourhood of the initial
+data giving rise to OSD collapse contains initial data cor-
+responding to collapse evolution giving rise to a singular-
+ity with the following property: one could trace outgoing
+past singular causal geodesics. This means that the end
+state of OSD collapse is unstable under small perturba-
+tions in initial data. Moreover, one can show the forma-
+tion of naked singularities (global and local) as an end
+state of gravitational collapse from suitable, physically
+reasonable initial data for various matter fields [13, 14].
+This implies that the initial conditions must be fine-tuned
+for the cosmic censorship conjecture to hold.
+In such a context, an important question one can ask
+here is as follows: what will be the end state of an un-
+hindered gravitational collapse of a fundamental matter
+field, such as a scalar field or a vector field, derived from
+an appropriate Lagrangian?
+The answer to this question has been achieved up to
+a certain extent. Scalar fields are fundamental matter
+fields derived from suitable Lagrangian.
+A real scalar
+field is a map defined on a smooth manifold as φ : M →
+R with a suitable continuity condition.
+Christodoulou
+showed that in the case of gravitational collapse of a
+massless scalar field φ (the scalar field Lagrangian is
+Lφ = (1/2)gµν∂µφ∂νφ), the set of initial data giving rise
+to a naked singularity as an end state has positive codi-
+mension in the entire initial data set [15, 16]. This means
+arXiv:2301.05083v1  [gr-qc]  12 Jan 2023
+
+2
+that the initial data set corresponding to naked singular-
+ity has a zero measure in the total initial data set. In
+other words, naked singularity in such cases is unstable
+under arbitrarily small perturbations in the initial data.
+One can have a massless scalar field with a poten-
+tial function Vs(φ) that still be a fundamental matter
+field.
+A massive scalar field will then be a particular
+case of a massless scalar field with a specific potential of
+the form Vs(φ) = (1/2)µ2φ2, where µ is the mass term.
+Goswami and Joshi [17] showed the example of the grav-
+itational collapse of a massless SH scalar field with a cer-
+tain potential Vs(φ) that ends up in a naked singularity.
+Mosani, Dey, Bhattacharya, and Joshi [18] conducted a
+similar investigation for a massless scalar field with a two-
+dimensional analogue of the Mexican hat-shaped Higgs
+field potential and found out that the end state of such
+unhindered scalar field collapse is a naked singularity.
+In addition to scalar fields as fundamental matter
+fields, vector fields are also fundamental matter fields de-
+rived from suitable matter Lagrangian. Geometrically,
+vector fields on a smooth manifold M can be thought of
+as sections on the tangent bundle π : TM → M, where
+π is a continuous surjection. A section is a smooth map
+σ : M → TM such that π ◦ σ is an identity map on
+M. From a particle physics point of view, the funda-
+mental nature of a vector field is different from that of a
+scalar field. There are many aspects, but one of the most
+important ones is that massive or massless vector fields
+mediate most particle physics processes. These represent
+the three fundamental interactions: quantum electrody-
+namics and weak and strong processes. A massless vector
+field with a potential function Vv(B) is again a funda-
+mental matter field. A massive vector field will then be
+a particular case of a massless vector field with a specific
+potential of the form Vv(B) = (1/2)µ2B, where µ is the
+mass term. Garfinkle, Mann, and Vuille [26] have studied
+the collapse of a massive vector field and numerically ob-
+tained the critical initial conditions. To our knowledge,
+much analytical work has not been done in investigating
+the causal structure of the end-state spacetime of the un-
+hindered gravitational collapse of matter fields that are
+vector fields.
+In this paper, in both the massless SH scalar field as
+well as vector field cases, we show that there are broad
+classes of potentials for which the configuration collapses
+and ends up in either a black hole or a naked singu-
+larity depending on the potential function chosen. We
+approach the causality investigation problem of scalar
+field as well as vector field collapse in a unified way, so
+to speak.
+As far as general relativity is concerned, it
+does not discriminate between whether a scalar field or
+a vector field seeds the matter field.
+The matter field
+is entirely identified by a rank two tensor field that we
+call the stress-energy tensor. As far as SH perfect fluid
+is concerned, one can identify a given matter field by
+the functional form of the equation of state parameter
+ω(a), where a is the scale factor of the collapsing cloud.
+We derive relevant equations of collapsing SH scalar field
+φ(a) and vector field ˜A(a) in the sub-sections of section
+II. The main body of section II contains discussions and
+relevant relations regarding the gravitational collapse of
+SH perfect fluids. In section III, we smoothly join the
+interior collapsing perfect fluid with an external gener-
+alized Vaidya spacetime. In section IV, we investigate
+the causal structure of the spacetime (condition of ob-
+taining a naked singularity) at the end of the collapse
+of the interior perfect fluid that is either a scalar field φ
+with potential Vs or a vector field ˜A with potential Vv.
+We also depict a few examples of well-known scalar fields
+and vector fields. In section V, we derive the criteria for
+the singularity, thus obtained in the end, to be strong of
+Tipler’s type. In the last section, we highlight the key
+points of the investigation. Here we use the geometrized
+units 8πG = c = 1 throughout.
+II.
+INTERIOR COLLAPSING MATTER FIELD
+Consider a gravitational collapse of a SH perfect fluid.
+The components of the stress-energy tensor in the coor-
+dinate basis {dxµ � ∂ν|0 ≤ µ, ν ≤ 3} of the comoving
+coordinates (t, x, y, z) are given by
+T µ
+ν = diag (−ρ, p, p, p) .
+(1)
+The spacetime geometry is governed by the flat (k = 0)
+Friedmann–Lemaˆıtre–Robertson–Walker (FLRW) metric
+ds2 = −dt2 + a2dΣ2,
+(2)
+where dΣ2 = dx2 + dy2 + dz2. Here a = a(t) is the scale
+factor such that a(0) = 1 and a(ts) = 0, where ts is the
+time of formation of the singularity. R = R(t, r) is the
+physical radius of the collapsing cloud and can be written
+as
+R(t, r) = ra(t),
+(3)
+where r is the radial spherical coordinate. For a FLRW
+spacetime Eq.(2), we have
+ρ = 3˙a2
+a2 ,
+(4)
+and
+p = −2¨a
+a − ˙a2
+a2 .
+(5)
+The overhead dot denotes the partial time derivative of
+a. Eq.(4) can be rewritten to obtain the dynamics of the
+collapse as
+˙a = −
+�
+ρ(a)
+3 a.
+(6)
+Differentiating the above equation once again gives us
+¨a = 1
+3a
+�aρ,a
+2
++ ρ
+�
+.
+(7)
+
+3
+Integrating Eq.(6), we obtain the time curve, which is
+t(a) =
+� 1
+a
+�3
+ρ
+da
+a .
+(8)
+The dynamics of the scale factor a(t) is, thus, the inverse
+of the LHS of the above equation. The time of formation
+of the singularity ts = t(0) is
+ts =
+� 1
+0
+�3
+ρ
+da
+a .
+(9)
+Now, let us consider a particular matter field ˆT from a
+set of all the possible SH perfect fluids. Choosing such
+an element means choosing a specific functional form of
+the equation of state parameter
+ω(a) = p
+ρ.
+(10)
+Using Eq.(4), Eq.(5), and Eq.(10), we can express the
+density of the matter field with the equation of state
+parameter ω as
+ρ = ρ0 exp
+�� 1
+a
+3 (1 + ω(a))
+a
+da
+�
+,
+(11)
+An SH perfect fluid is a fundamental matter field since
+it can be derived by a fundamental matter Lagrangian.
+In the following two subsections, we will describe two
+distinct ways of obtaining such a matter field.
+A.
+Scalar field collapse
+We prove that any SH perfect fluid is equivalent to a
+SH scalar field φ(a) with a suitable potential Vs(a), as
+far as the gravitational collapse is concerned. If φ(a) is
+invertible, then the following statement holds: Any SH
+perfect fluid is gravitationally equivalent to a SH scalar
+field φ with a suitable potential Vs(φ).
+Consider a real scalar field defined on the manifold M
+as
+φ : M → R.
+(12)
+The Lagrangian of a massless scalar field φ with potential
+Vs(a) is given by
+Lφ = 1
+2gµν∂µφ∂νφ − Vs(φ),
+(13)
+The stress-energy tensor is obtained from the Lagrangian
+Lφ as
+Tµν = −
+2
+√−g
+δ (√−gLφ)
+δgµν
+.
+(14)
+The density (ρs) and the isotropic pressure (ps) are sub-
+sequently expressed in terms of the time derivative of the
+scalar field and its potential as
+ρs = 1
+2
+˙φ2 + Vs
+(15)
+and
+ps = 1
+2
+˙φ2 − Vs.
+(16)
+The overhead dot denotes the time derivative of the func-
+tions.
+From Eq.(15) and Eq.(16), and from using the
+chain rule ˙φ = φ,a ˙a, we get
+ρs + ps = φ2
+,a ˙a2.
+(17)
+We now equate ρs = ρ and ps = p. Using Eq.(5) and
+(17), along with replacing ˙a and ¨a using Eq.(6) and (7),
+one obtains the expression of density as a function of a
+as
+ρs = ρ0 exp
+�� 1
+a
+aφ2
+,ada
+�
+.
+(18)
+From Eqs.(15) and (16), we get
+ps = ρs − 2Vs.
+(19)
+Using Eq.(6) in Eq.(17), we get
+ρs
+�
+1 − φ2
+,aa2
+3
+�
++ ps = 0.
+(20)
+Using Eqs.(19) and (20), we get
+Vs(φ) = ρs
+�
+1 − φ2
+,aa2
+6
+�
+.
+(21)
+Using Eq.(17), Eq.(6) in Eq.(5), one obtains
+ρs,a
+ρs
+= −φ,2
+a
+a .
+(22)
+We have, using Eq.(10), Eq.(15) and Eq.(16),
+Vs = ρs
+2 (1 − ω) .
+(23)
+Now from Eq.(21) and Eq.(23), we have
+φ(a),a = ±
+�
+3 (1 + ω(a))
+a
+.
+(24)
+Integrating the above equation, one obtains
+φ(a) = ±
+� 1
+a
+�
+3 (1 + ω(a))
+a
+da + c.
+(25)
+From Eq. (11) and Eq.(21) we have
+Vs(a) = ρ0
+�1 − ω(a)
+2
+�
+exp
+�� 1
+a
+3 (1 + ω(a))
+a
+da
+�
+.
+(26)
+Hence, we proved that given the functional form of
+
+4
+the equation of state parameter ω(a), one could obtain
+the corresponding scalar field φ(a) given by Eq.
+(25)
+with potential Vs(a) given by Eq. (26). As long as φ(a)
+is invertible (or, in other words, a bijective map from
+(0, 1] → R), we obtain a(φ), at least in principle, using
+which, we get Vs(φ).
+Alternatively, given a scalar field φ(a), one can obtain
+the corresponding perfect fluid ˆT (or the ω(a) by which
+it is identified), using Eq.(24).
+On the other hand, we can also start with a given scalar
+field potential V (φ). One can use Eq.(18) and Eq.(21) to
+obtain the ordinary nonlinear differential equation
+H
+�
+a, φ, dφ
+da , d2φ
+da2
+�
+= 0,
+(27)
+that can be solved in principle, to obtain φ(a), and later
+obtain ω(a) using Eq.(24). Hence, given a scalar field po-
+tential Vs(φ), one can obtain the corresponding ˆT (iden-
+tified by ω(a)) in the above manner.
+B.
+Vector field collapse
+We prove that any SH perfect fluid is equivalent to a
+SH vector field ˜A(a) with a suitable potential Vv(a), as far
+as the gravitational collapse is concerned. If B(a) is in-
+vertible, then the following statement holds: Any SH per-
+fect fluid is gravitationally equivalent to a SH vector field
+˜A with a suitable potential Vv(B) (where B = g( ˜A, ˜A)).
+Consider a vector field
+˜A : M → TM.
+(28)
+with potential V (B). For a fixed p ∈ M, ˜A(p) = Aµdxµ,
+where Aµ = (A0, Ai), 1 < i < 3 (in the comoving carte-
+sian coordinate basis). Here B = gαβAαAβ. We con-
+sider a SH pure vector field: A0 = 0 and Ai = A ∈ R
+��i ∈ (1, 2, 3). For such a vector field, B = 3A2/a2.
+The Lagrangian of a massless vector field ˜A with po-
+tential Vv(B) is given by
+L ˜
+A = −1
+4F µνFµν − Vv(B).
+(29)
+F is a two form called the field strength and can be writ-
+ten in terms of wedge product as F = Fµνdxµ ∧ dxν.
+The field strength is the exterior derivative of the vec-
+tor field ˜A, i.e.F = d ˜A. The components are written as
+Fµν = ∇µAν − ∇νAµ.
+The stress-energy tensor is obtained from the La-
+grangian L ˜
+A as
+Tµν = −
+2
+√−g
+δ (√−gL ˜
+A)
+δgµν
+.
+(30)
+This gives us
+Tµν = −1
+4FαβF αβgµν −Vv(B)gµν +FµαF
+α
+ν
++2V ′
+vAµAν.
+(31)
+The overhead prime denotes the ordinary derivative with
+respect to B. The density and the isotropic pressure are
+subsequently expressed in terms of the time derivative of
+the vector field component and its potential as
+ρv = 3
+2
+˙A2
+a2 + Vv(B),
+(32)
+and
+pv = 1
+2
+˙A2
+a2 − Vv(B) + 2V ′
+v
+A2
+a2 .
+(33)
+We now equate ρv = ρ and pv = p.
+From Eq.(32)
+and Eq.(4), we obtain
+Vv = ρv
+�
+1 − 1
+2A,2
+a
+�
+.
+(34)
+Substituting for ρ(a) from Eq.(11), we obtain
+Vv = ρ0 exp
+�� 1
+a
+3 (1 + ω(a))
+a
+da
+� �
+1 − 1
+2A,2
+a
+�
+(35)
+On differentiating Eq.(34) with respect to B we obtain,
+V ′
+v =
+ρv,a
+�
+1 − A,2
+a
+2
+�
+− ρvA,a A,aa
+6A2
+a2
+�
+A,a
+A − 1
+a
+�
+(36)
+Using Eq.(33), Eq.(4), and Eq.(5), we obtain
+aρv,a
+3
++ ρv
+�
+1 + 1
+6A,2
+a
+�
+= Vv − V ′
+v
+A2
+a2 .
+(37)
+Substituting for Vv and V ′
+v from Eq.(34) and Eq.(36),
+and also substituting for ρ,a (by differentiating Eq.(11))
+in Eq.(37), we obtain a second order nonlinear differential
+equation
+G
+�
+a, ω, A, dA
+da , d2A
+da2
+�
+= 0,
+(38)
+where G is
+G =d2A
+da2 − 4
+A
+�dA
+da
+�2
++ 1
+2a (5 − 3ω) dA
+da + 6
+A (1 + ω)
+− 3 (1 + ω)
+a
+�dA
+da
+�−1
+.
+(39)
+For a fixed ω(a), solving this differential equation with
+two initial conditions gives us A(a), and consequently,
+the vector field ˜A.
+Hence, we proved that given the functional form of the
+equation of state parameter ω(a), one could obtain the
+corresponding vector field ˜A using Eq.(38), and conse-
+quently, the vector field potential Vv(a) using Eq.(35).
+
+5
+SH Perfect Fluid
+Characterised by the equation of state 
+parameter 𝜔(a)
+SH Scalar Field
+Characterised by 𝜙(a)
+SH Vector Field
+Characterised by Ã(a)
+FIG. 1: A spatially homogeneous (SH) perfect fluid (governed by a flat FLRW spacetime metric) is completely
+characterized by the equation of state parameter ω(a), Eq.(10) of the matter field. This matter field is obtained
+from fundamental matter Lagrangian. Hence, the same matter field is also characterized by an SH scalar field φ(a),
+Eq.(25) or its potential Vs(a), Eq.(26) (Vs(φ) if φ(a) is invertible). Similarly, it can also be characterized by an SH
+vector field ˜A(a), Eq.(38) or its potential Vv(a), Eq.(35) (Vv(B) if B(a) is invertible). This schematic diagram
+depicts the equivalence between the gravitational collapse of SH Perfect fluid, Scalar field and Vector field. By
+spatial homogeneity, we mean homogeneous on a three-dimensional spacelike orbit with a six-dimensional isometry
+group G6 corresponding to the spacetime [3].
+Now, from the functional form A(a), we obtain B(a). As
+long as B(a) is invertible (or, in other words, a bijec-
+tive map from (0, 1] → R), we obtain a(B), at least in
+principle, using which, we get Vv(B).
+Alternatively, given a vector field ˜A(a), one can obtain
+the corresponding perfect fluid ˆT (or the ω(a) by which
+it is identified), using Eq.(38).
+On the other hand, we can also start with a given vec-
+tor field potential Vv(B). One can differentiate Eq.(35),
+and do some rearrangements to obtain
+ω(a, A, dA
+da , d2A
+da2 )
+as
+ω = 2AV ′
+aV
+�A
+a − dA
+da
+�
+−a
+3
+dA
+da
+d2A
+da2
+�
+1 − 1
+2
+�dA
+da
+�2�−1
+−1.
+(40)
+Substituting Eq.(40) in Eq.(38), we obtain
+˜G
+�
+a, A, dA
+da , d2A
+da2
+�
+= 0.
+(41)
+In principle, this differential equation can be solved to
+obtain A(a), which, when substituted in Eq.(40), gives
+us ω(a). Hence, given a vector field potential V (B), one
+can obtain the corresponding ˆT (identified by ω(a)) in
+the above manner.
+III.
+EXTERIOR GENERALIZED VAIDYA
+SPACETIME
+The collapsing vector field spacetime (g−
+µν) can be
+joined smoothly with the exterior generalized Vaidya
+spacetime (g+
+µν) so that their union forms a valid solution
+of the Einstein’s field equations. The interior FLRW and
+the exterior generalized Vaidya spacetime [19] are respec-
+tively given as
+ds2
+− = −dt2 + a(t)2dr2 + r2
+ba(t)2dΩ2,
+(42)
+and
+ds2
++ = −
+�
+1 − 2M(R, v)
+R
+�
+dv2 − 2dvdR + R2dΩ2. (43)
+Here, v is the retarded null coordinate, R is the general-
+ized Vaidya radius, and rb is the value of the radial co-
+ordinate r corresponding to the matching hypersurface,
+or in other words, the radial coordinate of the outermost
+shell of the collapsing scalar/vector field cloud. The mat-
+ter field corresponding to the generalized Vaidya space-
+time is a combination of Type I and type II, such that
+the components of the stress-energy tensor written in the
+
+6
+orthonormal basis appear as
+Tab =
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+¯ϵ
+2 + ϵ
+¯ϵ
+2
+0
+0
+¯ϵ
+2
+¯ϵ
+2 − ϵ 0
+0
+0
+0
+P
+0
+0
+0
+0 P.
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+(44)
+ϵ = P = 0 and ¯ϵ ̸= 0 corresponds the usual Vaidya
+spacetime as a special case. ¯ϵ = 0 and ϵ ̸= 0 corresponds
+to a sub-class of Type I matter field. The generalized
+Vaidya solution encompasses many known Einstein field
+equations solutions. Matching the first and second fun-
+damental forms for the interior and exterior metric on Σ
+gives the following equations:
+R(t) = R(t, rb) (= rba(t)) ,
+(45)
+F(t, rb) = 2M(R, v),
+(46)
+�dv
+dt
+�
+Σ
+=
+1 + ˙R
+1 − F (t,rb)
+R
+,
+(47)
+and
+M(R, v),R = F(t, rb)
+2R
++ R ¨R.
+(48)
+Here, F = R ˙R2 is the Misner-Sharp mass function of
+the collapsing spherical SH perfect fluid. Using the rela-
+tion (45), we can relate the generalized Vaidya mass with
+the density of the interior collapsing SH spherical perfect
+fluid cloud as
+M = ρ
+6R3.
+(49)
+Using Eq.(7), differentiation of Eq.(25) with respect to
+a, and Eq.(49), in Eq.(48) we get
+M,R = 3M
+R
+�
+1 + (1 + ω(a))r2
+b
+R2
+�
+,
+(50)
+integrating which we obtain
+M(R, v) = M1(v) exp
+��
+3
+R
+�
+1 + (1 + ˜ω (R))r2
+b
+R2
+�
+dR
+�
+.
+(51)
+Here M1(v) is a constant of integration and is a function
+of null coordinate v, and
+˜ω(R) = ω
+�R
+rb
+�
+.
+Eq.(51) gives us the expression of the generalized Vaidya
+mass function of the exterior generalized Vaidya space-
+time, in terms of interior collapsing perfect fluid equation
+of state parameter ω, to ensure smooth matching at the
+matching hypersurface.
+For the exterior matter field to satisfy the weak energy
+condition, ¯ϵ and ϵ should be non-negative [19].
+These
+inequalities, in turn, put restrictions on the generalized
+Vaidya mass function as
+M,v ≤ 0,
+and
+M,R ≥ 0.
+(52)
+Using Eq.(50) and Eq.(51) in the above two relations, we
+obtain
+M1,v ≤ 0,
+(53)
+and
+�1 + ω(a)
+a2
+�
+≥ 0.
+(54)
+The inequality (54) is always satisfied if the interior col-
+lapsing matter field obeys the weak energy condition.
+Hence, Eq.(53) is the only restriction on the generalized
+Vaidya mass function for the exterior spacetime to obey
+at least the weak energy condition.
+Now,
+we have a complete solution of Einstein’s
+field equations consisting of an interior collapsing SH
+scalar/vector field (with some potential) and the exte-
+rior generalized Vaidya solution, matched smoothly at
+the matching hypersurface. The free functions are cat-
+egorically the potential function (Vs(φ) in case of scalar
+field collapse, and Vv(B) in case of vector field collapse),
+and the component of generalized Vaidya mass function
+M1(v), the latter one restricted by the inequality (53).
+It is evident that the choice of M1(v) does not affect the
+causal structure of the spacetime obtained as an end-
+state of unhindered gravitational collapse.
+Of course,
+instead of considering the potential function Vs(φ) (or
+Vv(B)) as a free function, one could also consider any one
+of the remaining functions: ω(a), ρ(a), φ(a) (or A(a)),
+Vs(a) (or Vv(a)) as a free function, without any trouble.
+In the next section, we study the end state of this class of
+global dynamical spacetime identified by any one of the
+free functions.
+IV.
+CAUSAL STRUCTURE AND STRENGTH
+OF THE SINGULARITY
+Once the singularity is formed as an end state of grav-
+itational collapse of the interior scalar (vector) field with
+potential Vs(φ) (Vv(B)), one can investigate whether or
+not causal geodesics can escape the singularity. Addition-
+ally, one can investigate whether or not such singularity
+is gravitationally strong in the sense of Tipler. The fol-
+lowing two subsections discuss these two properties.
+
+7
+Massless scalar field
+Vs(φ) = 0
+φ(a) = c ±
+√
+6 log a
+strong
+BH
+Homogeneous dust (ω = 0)
+Vs(φ) ∝ exp
+�√
+3φ
+�
+φ(a) = c ±
+√
+3 log a
+strong
+BH
+Goswami/ Joshi [17] (ω = − 2
+3) (SF1)
+Vs(φ) ∝ exp φ
+φ(a) = c ± log a
+strong
+NS
+Two dimensional analog of Mexican hat [18]
+(SF2)
+Vs(φ) = 1
+2µφ2 + λφ4
+φ(a) = ±2
+√
+2√c − log a
+weak
+NS
+TABLE I: Four examples of spatially homogeneous scalar fields that collapse to form a singularity that is either
+hidden (blackhole or BH) or (naked singularity or NS). In the fourth example, µ = − 16
+3 λ. The first three types end
+up in a strong singularity in the sense of Tipler.
+Massless vector field
+Vv(B) = 0
+strong
+BH
+Massive vector field
+Vv(B) = − 1
+2µ2B
+strong
+BH
+VF1
+Vv(a) as in Fig.(3)
+strong
+NS
+VF2
+Vv(a) as in Fig.(3)
+weak
+NS
+TABLE II: Four examples of spatially homogeneous vector fields that collapse to form a singularity that is either
+hidden within a black hole (BH) or is naked (NS). The ones mentioned in the third and the fourth row are newly
+constructed vector fields from known scalar fields (mentioned in the third [17] and the fourth [18] row of Table 1,
+respectively) by exploiting the gravitational equivalence depicted in Fig.(1). The corresponding vector field
+component A(a) for each case is plotted in Fig.(2-3). The first three types end up in a gravitationally strong
+singularity in the sense of Tipler.
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+-0.5
+0.0
+0.5
+1.0
+a
+A
+a
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0
+2
+4
+6
+a
+V
+FIG. 2: The dynamics of the vector field component A(a) in the case of the massive (µ = 1) vector field ˜A (Left
+panel) and its potential Vv(a) (Right panel). First we obtain ω(a, A, dA
+da , d2A
+da2 ) by substituting Vv(B) = − 1
+2µ2B in
+Eq.(40). Substituting for ω(a, A, dA
+da , d2A
+da2 ) in Eq.(41) and solving the differential equation with initial conditions
+A(1) = 1 and A′(1) = 2, we obtain A(a). Consequently, substituting Vv(B) = − 1
+2µ2B, and the obtained A(a) in
+Eq.(40), we obtain ω(a), Further substitution of ω(a) in Eq.(35), we obtain Vv(a).
+
+8
+0.2
+0.4
+0.6
+0.8
+1.0
+-3
+-2
+-1
+0
+1
+2
+3
+a
+Vv
+(a)
+0.0
+0.5
+1.0
+1.5
+2.0
+-10
+-8
+-6
+-4
+-2
+0
+2
+4
+B
+Vv
+(b)
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+-300
+-200
+-100
+0
+100
+200
+a
+Vv
+(c)
+0.0
+0.5
+1.0
+1.5
+2.0
+-800
+-600
+-400
+-200
+0
+200
+B
+Vv
+(d)
+A1
+A2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+a
+A
+(e)
+FIG. 3: (a) and (c): Vector field potentials Vv(a) corresponding to newly constructed vector fields VF1 (orange)
+and VF2 (green), as mentioned in the third and fourth row of the Table (II), respectively. (b) and (d): The same
+vector field potentials Vv(B) as function of B. (e): The vector field components A(a) in both of these cases. In the
+latter example, µ = −8/3 and λ = 1. First, we obtain ωi(a) , using Eq.(25) (Here i ∈ 1, 2 corresponds to VF1 and
+VF2 respectively). Then we obtain the vector field components Ai(a) by solving the differential Eq.(38) with initial
+conditions Ai(1) = 1 and A′
+i(1) = 10. Further substitution of ωi(a) and the obtained Ai(a) in Eq.(35), we get
+Vv(i)(a). Once Vv(i)(a) is obtained, we obtain Vv(i)(B).
+A.
+Causal structure of the singularity
+We say that a singularity formed due to unhindered
+gravitational collapse is naked if there exists a family of
+outgoing causal curves whose past endpoint is the singu-
+larity. In the future, these curves can either reach a far-
+away observer or fall back to the singularity. The singu-
+larities are then termed globally naked and locally naked,
+respectively. Whether or not the singularity is naked es-
+sentially depends on the geometry of trapped surfaces as
+the collapse evolves. Trapped surfaces are two-surfaces in
+the spacetime on which not only the ingoing congruence
+but also the outgoing congruence necessarily converge.
+Convergence or otherwise of the outgoing null geodesic
+congruence is determined by the behaviour of its expan-
+sion scalar, which we denote here as θl (t, r). It is ex-
+pressed in terms of the metric coefficients, in comoving
+spherical coordinates as,
+θl = 2
+R
+�
+1 −
+�
+ρR2
+3
+�
+.
+(55)
+
+9
+EH
+AH
+0.0
+0.1
+0.2
+0.3
+0.4
+R
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+t
+(a) Massless scalar field (Vs = 0)
+EH
+AH
+0.0
+0.1
+0.2
+0.3
+0.4
+R
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+0.35
+t
+(b) Massless vector field (Vv = 0)
+0.0
+0.1
+0.2
+0.3
+0.4
+R
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+t
+(c) SF1/VF1
+0.0
+0.1
+0.2
+0.3
+0.4
+R
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+t
+(d) SF2/VF2
+FIG. 4: Spacetime diagram of the examples of spatially homogeneous scalar fields and vector fields mentioned in
+Tables I and II. The solid black curve in each of them represents the boundary of the collapsing cloud. Upper panel:
+The singularity is not visible in both examples. Lower Panel: In the case of SF1/VF1, the singularity forms in a
+finite comoving time and is globally visible because of the absence of the apparent and event horizons. In the case of
+SF2/VF2, the singularity forms in an infinite comoving time. However, an ultra-high density region is obtained in
+finite comoving time, which can be visible globally because of the absence of the apparent and event horizons.
+The region in which θl < 0 is called the trapped region.
+The boundary of the trapped region, given by θl = 0, is
+called the apparent horizon. If the neighbourhood of the
+singular center is surrounded by a trapped region since
+before the time of formation of the singularity ts, then
+it is covered, and we get a black hole. Hence, the nec-
+essary condition for singular null geodesic congruence to
+escape the singularity is the absence of a trapped region,
+which is ensured by the condition θl(ts, r) > 0 for such
+congruence. The absence of trapped region in the neigh-
+bourhood of the singularity (t, r) = (ts, 0) is ensured by
+the following inequality:
+lim
+t→ts
+ρR2
+3
+≤ lim
+a→0
+ρ(a)r2
+ba2
+3
+< 1,
+(56)
+The inequality (56) is definitely satisfied if
+lim
+a→0 ρ(a) < 1
+a2 .
+(57)
+For
+lim
+a→0 ρ(a) = k
+a2 ,
+for some k ∈ R+, the inequality is satisfied only for
+rb <
+�
+3/k. Rewriting the inequality (57) in terms of
+the equation of state parameter ω(a) using Eq.(11), one
+obtains
+lim
+a→0 ρ0a2 exp
+�� 1
+a
+3 (1 + ω(a))
+a
+�
+da < 1.
+(58)
+If a collapsing matter field with the equation of state
+parameter ω(a) satisfies the inequality (58), then it will
+
+10
+up in a naked singularity [17]. In the case of otherwise,
+the final outcome is a black hole.
+Hence, as decided by the above inequality, we get a
+class of SH matter fields that include scalar and vector
+fields, identified by the functional form ω(a), that goes
+to either the blackhole or naked singularity final state as
+an end state of unhindered gravitational collapse.
+In the case of scalar field collapse, the restriction (58)
+on ω(a) gives us a restriction on the scalar field φ(a) us-
+ing Eq.(25), and the scalar field potential function Vs(a)
+using Eq.(26). Hence, obtaining a class of ω(a) is grav-
+itationally equivalent to obtaining a class of scalar field
+potentials Vs(a) that goes to the naked singularity as
+an end state of unhindered gravitational collapse. More-
+over, suppose φ(a) is a bijective map from (0, 1] → R.
+In that case, obtaining a class of ω(a) is gravitationally
+equivalent to obtaining a class of scalar field potentials
+Vs(φ) = Vs(a(φ)) that goes to the naked singularity as
+an end state of gravitational collapse.
+Similarly, in the case of vector field collapse, the re-
+striction (58) on ω(a) gives us a restriction on the vector
+field ˜A (or more specifically, a restriction on the vec-
+tor field component A(a)) obtained by solving the dif-
+ferential Eq.(38), and the vector field potential function
+Vv(a) obtained by substituting A(a) and ρ from Eq.(11),
+in Eq.(34).
+Hence, obtaining a class of ω(a) is gravi-
+tationally equivalent to obtaining a class of vector field
+potential Vv(a) that goes to the naked singularity as an
+end state of unhindered gravitational collapse.
+More-
+over, suppose A(a) is a bijective map from (0, 1] → R.
+In that case, obtaining a class of ω(a) is gravitationally
+equivalent to obtaining a class of vector field potential
+Vv(A) = Vv(a(A)) that goes to the naked singularity as
+an end state of unhindered gravitational collapse.
+In Table (I) and (II), we discuss examples of such
+scalar field collapse and vector field collapse that ends
+up in either a black hole or a naked singularity.
+Ex-
+ploiting the equivalence between SH perfect fluids, scalar
+fields with potential Vs(a), and vector fields with poten-
+tial Vv(a), we construct two examples of collapsing vector
+fields with potential out-of-known examples of collapsing
+scalar fields with potentials, giving rise to the naked sin-
+gularity as an end state.
+The first example of a collapsing vector field with po-
+tential Vv(a) is constructed from the scalar field with po-
+tential mentioned in the third row of Table (I) [17]. The
+perfect fluid corresponding to such scalar field example
+has an equation of state parameter ω(a) = − 2
+3. The con-
+structed collapsing vector field ˜A = (0, A, A, A) (in the
+comoving coordinate basis) has the property (dynamics
+of A(a) and Vv(a)) as shown in Fig.(3). Refer to the third
+row of Table (II).
+The second example of a collapsing vector field with
+potential Vv(a) is constructed from the scalar field with
+potential mentioned in the fourth row of Table (I) [18].
+Such a scalar field has a two-dimensional analogue of
+Mexican hat-shaped potential. The constructed collaps-
+ing vector field ˜A = (0, A, A, A) (in the comoving co-
+ordinate basis) has the property (dynamics of A(a) and
+Vv(a)) as shown in Fig.(3). Refer to the fourth row of
+Table (II). The spacetime diagrams of some of the ex-
+amples in Table (I) and (II) are plotted in Fig.(4).
+B.
+Strength of the singularity
+Generally, a singularity in the spacetime manifold is
+identified by the existence of at least one past/future in-
+complete geodesic. However, in the case of singularities
+forming as the end state of a gravitational collapse, apart
+from the existence of such incomplete geodesics, one ex-
+pects an additional physical property as follows: An ob-
+ject approaching such singularity should be crushed to
+zero volume. We call such a singularity gravitationally
+strong in the sense of Tipler [20]. A precise definition of
+a strong singularity is as follows:
+Consider a smooth spacetime manifold (M, g) and a
+causal geodesic γ : [t0, 0) → M. Let λ be an affine pa-
+rameter along this geodesic. Let ξ(i), (0 ≤ i ≤ 2 in the
+case of null geodesic, 0 ≤ i ≤ 3 in the case of timelike
+geodesic) be the independent Jacobi vector fields. The
+wedge product of these Jacobi fields gives us the volume
+form V = � ξ(i). We say that a singularity is gravita-
+tionally strong in the sense of Tipler if this volume form
+vanishes as λ → 0.
+Clarke and Krolak [21] related the existence of a Tipler
+strong singularity with the growth rate of the curvature
+terms as follows: At least along one null geodesic with
+affine parameter λ (such that λ → 0 as the singularity is
+approached), the following inequality
+lim
+λ→0 λ2RijKiKj > 0
+(59)
+should hold for the singularity to be strong in the sense of
+Tipler. Here Ki = dxi
+dλ are the tangents to the chosen null
+geodesic, and xi is the coordinate system. This condition
+puts a lower bound on the growth of the curvature scalar.
+In the spherical coordinate system (t, r, θ, φ), the radial
+null geodesic equation reads
+dt
+dr = a.
+(60)
+Hence, we have the relation between the tangents Kt and
+Kr as
+Kt = aKr,
+(61)
+and subsequently, in terms of the affine parameter,
+Kt = R
+λ ,
+and
+Kr = r
+λ.
+(62)
+The inequality (59) can then be written in terms of ω as
+lim
+a→0
+�
+r2(1 + ω)ρ0 exp
+�� 1
+a
+3(1 + ω)
+a
+da
+��
+> 0
+(63)
+
+11
+Hence, the singularity formed due to the gravitational
+collapse of a scalar/vector field is strong in the sense of
+Tipler if the following inequality holds (assuming that
+the weak energy condition is respected):
+lim
+a→0 exp
+�� 1
+a
+3(1 + ω)
+a
+da
+�
+> 0.
+(64)
+Hence, (along with using the condition (58)) one can ob-
+tain a naked singularity that is strong in the sense of
+Tipler for that ω that satisfies the following constraint:
+0 < lim
+a→0 exp
+�� 1
+a
+3(1 + ω)
+a
+da
+�
+< O(a−2).
+(65)
+This constraint gives us the class of SH collapsing mat-
+ter fields that we identify by ω(a), which ends up in
+strong curvature naked singularity. Or in other words,
+we have a class of scalar/vector field potentials corre-
+sponding to the given scalar/vector field that collapses
+to a strong naked singularity.
+As an example, in Ta-
+bles (I) and (II), we mention the causal property and
+the strength of the singularity formed due to the gravi-
+tational collapse of four different scalar/vector fields.
+V.
+CONCLUSIONS AND REMARKS
+Following are the concluding remarks:
+1. Unlike the singularity theorems that provide rig-
+orous proof of the existence of incomplete causal
+geodesics under rather generic conditions, one does
+not currently have proof or disproof of the cosmic
+censorship hypothesis.
+In fact, we need a math-
+ematically rigorous formulation of this conjecture,
+which is not available currently, before we can prove
+or disprove it.
+Under the situation at present, we can only spec-
+ulate its validity or otherwise.
+Proposed coun-
+terexamples, hence have great importance in under-
+standing whether naked singularities, in fact, exist
+or not in our universe. Through such analysis of
+gravitational collapse models only, one could pos-
+sibly hope to arrive at a suitable formulation of
+cosmic censorship. The collapse of inhomogeneous
+dust and the Vaidya null fluids were the first exam-
+ples proposed to produce naked singularities. How-
+ever, an important objection could be that, even if
+astrophysically interesting, they are not fundamen-
+tal forms of matter [7, 22].
+One could then ask
+whether the collapse of matter configuration that
+is obtained from a fundamental matter Lagrangian
+ends up in a naked singularity. Scalar fields with
+potential and vector fields with potential are fun-
+damental matter fields in this sense. Here we show
+that not just one particular choice of these fields
+but an entire class of such types could collapse and
+form a naked singularity as an end state. This basi-
+cally divides the allowed class of potential functions
+into classes that take the unhindered collapse to a
+black hole or naked singularity.
+2. To achieve this, we show equivalence between SH
+(a) Perfect fluid: characterized by ω(a),
+(b) Massless scalar field φ: characterized by φ(a)
+or its potential Vs(a) or Vs(φ) (if φ(a) is in-
+vertible), and
+(c) Massless vector field
+˜A:
+characterized by
+A(a), or its potential Vv(a), or Vv(B) (if B(a)
+is invertible).
+as far as the gravitational collapse is concerned.
+This gravitational equivalence is described in sub-
+sections of section (II) and depicted in Fig.(1).
+Now, if the functional form of ω(a) satisfies the
+inequality (58), then the singular null geodesic con-
+gruence, if at all there exists, does not get trapped
+as a → 0.
+Hence, we have a class of functions
+ω(a) corresponding to a naked singularity as an end
+state of gravitational collapse. Now, because of the
+above equivalence, in the case of an SH scalar field
+collapse, one then has a class of scalar field func-
+tion φ(a), or a class of scalar field potential Vs(a),
+or a class of scalar field potential in terms of φ,
+i.e. Vs(φ) (provided φ(a) is invertible), that corre-
+sponds to the naked singularity as an end state.
+Similarly, in the case of an SH vector field col-
+lapse, one has a class of vector field component
+function A(a), or a class of vector field potential
+Vv(a), or a class of vector field potential in terms
+of B = g( ˜A, ˜A), i.e. Vs(B) (provided B(a) is in-
+vertible), that corresponds to the naked singularity
+as an end state.
+3. A naked singularity formed due to gravitational col-
+lapse may or may not be relevant if they are not
+gravitationally strong in the sense of Tipler [20].
+Here, we show a class of ω(a) that satisfies the in-
+equalities (65) that corresponds to the formation
+of a strong curvature naked singularity. Using ar-
+guments similar to point no. 2 of this section, we
+have equivalently shown a class of scalar field po-
+tential (in case of scalar field collapse) and a class
+of vector field potential (in case of vector field col-
+lapse) that corresponds to a strong curvature naked
+singularity.
+4. For the sake of completion, we study the global
+spacetime, consisting of the interior collapsing
+scalar/vector field and the exterior generalized
+Vaidya spacetime. The smooth matching demands
+a restriction on the free function, that is, the gener-
+alized Vaidya mass function, in terms of the prop-
+erty of the interior collapsing scalar/vector field.
+
+12
+We have fulfilled this demand by deriving the ex-
+pression of the generalized Vaidya mass in terms
+of the equation of state parameter of the interior
+collapsing field in Eq.(51).
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+Archive 56, 455 (1939).
+[2] S. Datt, Zs. f. Phys. 108 314 (1938).
+[3] G.F.R. Ellis, S.T.C. Siklos and J. Wainwrighit, in Dy-
+namical systems in cosmology, Eds. J. Wainwright and
+G.F.R. Ellis, (Cambridge University Press, Cambridge,
+England, 1997).
+[4] R. Penrose, Riv. Nuovo Cimento Soc. Ital. Fis. 1, 252
+(1969).
+[5] R. Geroch, Journal of Mathematical Physics, 11, 2, 437-
+449 (1970).
+[6] S. W. Hawking and G. F. R. Ellis, The large scale struc-
+ture of spacetime, Cambridge University Press (1973).
+[7] P. S. Joshi, Global Aspects in Gravitation and Cosmology
+(Clendron Press, Oxford, 1993).
+[8] R. Geroch and G. Horowitz, ‘Global structure of space-
+times’, in General Relativity:
+An Einstein Centenary
+Survey, eds S. W. Hawking and W. Israel, Cambridge:
+Cambridge University Press (1979).
+[9] S. W. Hawking and W.Israel, ‘An introductory survey’, in
+General Relativity: An Einstein Centenary Survey, eds
+S. W. Hawking and W. Israel. Cambridge: Cambridge
+University Press (1979).
+[10] R. Penrose, ‘Singularities and time asymmetry’, in Gen-
+eral Relativity: An Einstein Centenary Survey, eds S. W.
+Hawking and W. Israel. Cambridge: Cambridge Univer-
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+[11] R. Penrose, Phys. Rev. Lett. 14, 57 (1965).
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+(2011).
+[13] P. S. Joshi, Gravitational Collapse, and Spacetime Singu-
+larities, (Cambridge University Press, Cambridge, Eng-
+land, 2007).
+[14] K. Mosani, D. Dey, P. S. Joshi, Phys. Rev. D 102,
+044037.
+[15] D. Christodoulou, Annals of Mathematics Annals of
+Mathematics, 140, 607 (1994).
+[16] D. Christodoulou, Annals of Mathematics, 149, 183
+(1999).
+[17] R. Goswami and P. S. Joshi, Modern Physics Letters A,
+22, 01, pp. 65-74 (2007).
+[18] Karim Mosani, Dipanjan Dey, Kaushik Bhattacharya
+and Pankaj S. Joshi, Phys. Rev. D 105, 064048 (2022).
+[19] A. Wang and Y. Wu, Gen. Relativ. Gravit. 31, 107
+(1999).
+[20] F. J. Tipler, Phys. Lett. 64A, 8 (1977).
+[21] C. J. S. Clarke and A. Krolak, J. Geom. Phys. 2, 127
+(1985).
+[22] D. M. Eardley, in ’Gravitation in Astrophysics’, ed. B.
+Carter and J. B. Hartle (Plenum, New York, 1987).
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+Rev. D 101, 044052 (2020).
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+337-361 (1986).
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+

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+arXiv:2301.02159v1  [math.NA]  5 Jan 2023
+Finite element approximation of scalar curvature in arbitrary
+dimension
+Evan S. Gawlik∗
+Michael Neunteufel†
+Abstract
+We analyze finite element discretizations of scalar curvature in dimension N ≥ 2.
+Our
+analysis focuses on piecewise polynomial interpolants of a smooth Riemannian metric g on a
+simplicial triangulation of a polyhedral domain Ω ⊂ RN having maximum element diameter h.
+We show that if such an interpolant gh has polynomial degree r ≥ 0 and possesses single-valued
+tangential-tangential components on codimension-1 simplices, then it admits a natural notion
+of (densitized) scalar curvature that converges in the H−2(Ω)-norm to the (densitized) scalar
+curvature of g at a rate of O(hr+1) as h → 0, provided that either N = 2 or r ≥ 1. As a special
+case, our result implies the convergence in H−2(Ω) of the widely used “angle defect” approxima-
+tion of Gaussian curvature on two-dimensional triangulations, without stringent assumptions on
+the interpolated metric gh. We present numerical experiments that indicate that our analytical
+estimates are sharp.
+1
+Introduction
+Many partial differential equations that arise in mathematical physics and geometric analysis involve
+the Riemann curvature tensor and its contractions.
+The scalar curvature R, which is obtained
+from two contractions of the Riemann curvature tensor, is particularly important; it serves as the
+integrand in the Einstein-Hilbert functional from general relativity, and it appears in the governing
+equation for two-dimensional Ricci flow. To approximate solutions to PDEs involving the scalar
+curvature, it is necessary to discretize the nonlinear differential operator that sends a Riemannian
+metric tensor to its scalar curvature.
+The goal of this paper is to construct and analyze such
+discretizations in arbitrary dimension N ≥ 2.
+We are specifically interested in the setting where a smooth Riemannian metric tensor g on
+a polyhedral domain Ω ⊂ RN is approximated by a piecewise polynomial Regge metric gh on a
+simplicial triangulation T of Ω having maximum element diameter h. Here, a metric is called a
+Regge metric on T if it is piecewise smooth and its tangential-tangential components are single-
+valued on every codimension-1 simplex in T . When such a metric is piecewise polynomial, it belongs
+to a finite element space called the Regge finite element space [11, 12, 21]. Regge metrics are not
+classically differentiable, so our first task will be to assign meaning to the scalar curvature of gh. Our
+definition, which is a natural generalization of one that is now well-established in dimension N = 2,
+treats the scalar curvature of gh as a distribution and regards it as an approximation of the densitized
+scalar curvature of g, i.e. the scalar curvature R times the volume form ω. For piecewise constant
+Regge metrics, our definition reduces to the classical definition of the distributional densitized
+∗Department of Mathematics, University of Hawai‘i at Manoa, Honolulu, HI, 96822, USA, egawlik@hawaii.edu
+†Institute for Analysis and Scientific Computing, TU Wien, Wiedner Hauptstr. 8-10, 1040 Wien, Austria,
+michael.neunteufel@tuwien.ac.at
+1
+
+curvature on piecewise flat spaces [8, 23]. It is a linear combination of Dirac delta distributions
+supported on (N − 2)-simplices S, weighted by the angle defect at S: 2π minus the sum of the
+dihedral angles incident at S. For piecewise polynomial Regge metrics of higher degree, it includes
+additional contributions involving the scalar curvature in the interior of each N-simplex and the
+jump in the mean curvature across each (N − 1)-simplex.
+We study the convergence of the distributional densitized scalar curvature of gh to the densitized
+scalar curvature of g under refinement of the triangulation. We show in Theorem 4.1 that in the
+H−2(Ω)-norm, this convergence takes place at a rate of O(hr+1) when gh is an optimal-order
+interpolant of g that is piecewise polynomial of degree r ≥ 0, provided that either N = 2 or r ≥ 1.
+Our numerical experiments in Section 5 suggest that these estimates are sharp in general.
+To put this convergence result into context, let us summarize some existing convergence results
+in the literature on finite element approximation of the scalar curvature. We first need to assemble
+some notation.
+Notation.
+In what follows, W s,p(Ω) denotes the Sobolev-Slobodeckij space of differentiability
+index s ∈ [0, ∞) and integrability index p ∈ [1, ∞], and ∥ · ∥W s,p(Ω) and | · |W s,p(Ω) denote the
+associated norm and semi-norm, which we always take with respect to the Euclidean metric. We
+denote Lp(Ω) = W 0,p(Ω) and Hs(Ω) = W s,2(Ω). For k ∈ N, we denote H−k(Ω) = (Hk
+0 (Ω))′, where
+Hk
+0 (Ω) denotes the space of functions in Hk(Ω) whose derivatives of order 0 through k − 1 have
+vanishing trace on ∂Ω, and the prime denotes the dual space. Occasionally we use weighted Lp and
+H−k spaces associated with a Riemannian metric g, which we denote by Lp(Ω, g) and H−k(Ω, g);
+see Section 4 and [16, Equation 4.1] for details.
+If g is a smooth Riemannian metric and gh is a Regge metric, then R(g) denotes the scalar
+curvature of g, (Rω)(g) denotes the densitized scalar curvature of g, (Rω)dist(gh) denotes the
+distributional densitized scalar curvature of gh (defined below in Definition 3.1), and R(q)
+h (gh)
+denotes the L2(Ω, gh)-projection of (Rω)dist(gh) onto the Lagrange finite element space of degree q.
+We also use the terms optimal-order interpolant, canonical interpolant, and geodesic interpolant
+below. The first of these is a catch-all term for any piecewise polynomial interpolant gh of g that
+belongs to the Regge finite element space and enjoys error estimates of optimal order in W s,p(T)-
+norms on N-simplices T; see Definition 4.2. The canonical interpolant is a specific interpolant
+(which is optimal-order) detailed in [21, Chapter 2]. The geodesic interpolant of g is the unique
+piecewise constant Regge metric gh with the property that the length of every edge in T , as
+measured by gh, agrees with the geodesic distance between the corresponding vertices in T , as
+measured by g.
+Summary of existing results.
+We can now summarize some existing results about the approx-
+imation of g’s curvature by gh’s distributional curvature. Throughout what follows, the letter r
+denotes the polynomial degree of gh.
+1. Cheeger, M¨uller, and Schrader [8, Equation (5.7) and Theorem 5.1] proved that if r = 0 and
+gh is the geodesic interpolant of g, then (Rω)dist(gh) converges to (Rω)(g) in the (setwise)
+sense of measures at a rate of O(h) in dimension N = 2 and O(h1/2) in dimension N ≥ 3.
+2. Gawlik [16, Theorem 4.1] proved that if r ≥ 1, N = 2, and gh is any optimal-order interpolant
+of g, then R(q)
+h (gh) converges to R(g) at a rate of O(hr) in the H−1(Ω, g)-norm and at a rate of
+O(hr−k−1) in the broken Hk(Ω)-norm, k = 0, 1, 2, . . . , r − 2, provided that q ≥ max{1, r − 2}.
+3. Berchenko-Kogan and Gawlik [4, Corollary 6.2] proved that if r ≥ 1, N = 2, and gh is any
+optimal-order interpolant of g, then (Rω)dist(gh) converges to (Rω)(g) at a rate of O(hr) in
+2
+
+the norm ∥u∥V ′,h = supv∈V,v̸=0⟨u, v⟩V ′,V /∥v∥V,h, where
+V = {v ∈ H1
+0(Ω) | v|T ∈ H2(T) ∀T ∈ T N}
+(1)
+and ∥v∥V,h = |v|H1(Ω) +
+��
+T∈T N h2
+T |v|2
+H2(T)
+�1/2
+. Here, hT denotes the diameter of T, and
+T N denotes the set of N-simplices in T .
+4. Gopalakrishnan, Neunteufel, Sch¨oberl, and Wardetzky [19, Theorem 6.5 and Corollary 6.6]
+proved that if r ≥ 0, N = 2, and gh is the canonical interpolant of g, then R(r+1)
+h
+(gh) converges
+to R(g) at a rate of O(hr+1) in the H−1(Ω, g)-norm and at a rate of O(hr−k) in the broken
+Hk(Ω)-norm, k = 0, 1, 2, . . . , r − 1.
+New results.
+As one can see from above, our analysis in this paper covers two important cases
+that have not yet been addressed in the literature:
+1. We prove a convergence result in the case where N ≥ 3 and r ≥ 1. This opens the door
+to the use of piecewise polynomial Regge metrics to approximate scalar curvature in high
+dimensions.
+2. We prove a convergence result in the case where N = 2, r = 0, and gh is an arbitrary
+optimal-order interpolant of g. This has been a longstanding gap in the literature on Gaussian
+curvature approximation. Previous efforts to address the case where N = 2 and r = 0 have
+relied on subtle properties of the geodesic interpolant [8] and the canonical interpolant [19].
+Our results establish the validity of Gaussian curvature approximations involving the angle
+defect without stringent assumptions on the interpolated metric tensor gh.
+Note that our analysis predicts no convergence at all in the H−2(Ω)-norm when N ≥ 3 and r = 0.
+Our numerical experiments suggest that this result is sharp for general optimal-order interpolants.
+However, for the canonical interpolant, numerical experiments suggest that (Rω)dist(gh) converges
+to (Rω)(g) in the H−2(Ω)-norm at a rate of O(h) when N ≥ 3 and r = 0. We intend to study this
+superconvergence phenomenon exhibited by the canonical interpolant in future work.
+Structure of the paper.
+Our strategy for proving convergence of (Rω)dist(gh) to (Rω)(g) con-
+sists of two steps. First, in Sections 2-3, we study the evolution of (Rω)dist(gh) under deformations
+of the metric, leading to an integral formula for the error (Rω)dist(gh) − (Rω)(g) which reads
+⟨(Rω)dist(gh) − (Rω)(g), v⟩V ′,V =
+� 1
+0
+bh(�g(t); σ, v) − ah(�g(t); σ, v) dt,
+∀v ∈ V.
+(2)
+Here, �g(t) = (1 − t)g + tgh, σ =
+∂
+∂t�g(t) = gh − g, V is the space defined in (1), and bh(�g(t); ·, ·)
+and ah(�g(t); ·, ·) are certain metric-dependent bilinear forms. In Section 4, we use techniques from
+finite element theory to estimate the right-hand side of (2), leading to Theorem 4.1.
+The approach above is similar to the one used in dimension N = 2 in [4, 16, 19], but there are
+a few important differences. First, we work with an integral formula for the error (Rω)dist(gh) −
+(Rω)(g) rather than an integral formula for the curvature itself. Previous analyses in [4, 16, 19]
+hinged on formulas of the latter type. Loosely speaking, in this paper we compute the evolution of
+the error along a one-parameter family of Regge metrics starting at g and ending at gh, whereas
+the papers [4, 16, 19] compute the evolution of the curvature along a pair of one-parameter families
+of metrics: one family that starts at the Euclidean metric δ and ends at gh, and one that starts at
+3
+
+δ and ends at g. The approach based on evolving the error appears to be better suited for proving
+optimal error estimates.
+Another key aspect of our analysis is our use of the H−2(Ω)-norm to measure the error. This
+norm is weaker than the ones used in [4, 16, 19], and it appears to be more natural for measuring
+the error in the curvature. For example, for piecewise constant Regge metrics in dimension N = 2,
+we show that convergence of (Rω)dist(gh) to (Rω)(g) holds in the H−2(Ω)-norm for any optimal-
+order interpolant of g, but numerical experiments suggest that it fails to hold in stronger norms
+when gh is not the canonical interpolant of g. A key tool that we use to prove convergence in
+H−2(Ω) is the near-equivalence of a certain pair of metric-dependent, mesh-dependent norms on
+V ; see Proposition 4.5. This equivalence is similar to one that Walker [27, Theorems 4.1 and 4.3]
+proved for an analogous family of mesh-dependent norms on triangulated surfaces.
+Additional comments.
+The formula (2) is not only useful for the error analysis, but it is also
+interesting in its own right. It has a differential counterpart (see Theorem 3.6) that reads
+d
+dt⟨(Rω)dist(�g(t)), v⟩V ′,V = bh(�g(t); σ, v) − ah(�g(t); σ, v),
+∀v ∈ V,
+(3)
+which mimics the formula
+d
+dt
+�
+Ω
+Rvω =
+�
+Ω
+(div div Sσ)vω −
+�
+Ω
+⟨G, σ⟩vω,
+∀v ∈ V
+(4)
+that holds for a family of smooth Riemannian metrics g(t) with densitized scalar curvature Rω and
+Einstein tensor G = Ric − 1
+2Rg. Here, Sσ = σ−g Tr σ, and div is the covariant divergence operator;
+see below for more notational details.
+The correspondence between (3) and (4) becomes even more transparent when one inspects the
+formulas for bh and ah (see Theorem 3.6). The bilinear form bh(�g; ·, ·) is (up to the appearance of
+S) a non-Euclidean, N-dimensional generalization of a bilinear form that appears in the Hellan-
+Herrmann-Johnson finite element method [1–3, 5–7, 9, 22].
+It can be regarded as the integral
+of div div Sσ against v, where div div is interpreted in a distributional sense. This link with the
+Hellan-Herrmann-Johnson method has previously been noted and used in dimension N = 2 [4, 16,
+19].
+The bilinear form ah(�g; ·, ·), which is only nonzero in dimension N ≥ 3, appears to play the role
+of
+�
+Ω⟨G, σ⟩vω, which is also only nonzero in dimension N ≥ 3. It gives rise to a natural way of
+defining the Einstein tensor in a distributional sense for Regge metrics. We discuss this more in
+Section 3.2. Among other things, we point out that the formula for ah contains a term involving
+the jump in the trace-reversed second fundamental form across codimension-1 simplices; the same
+quantity arises in studies of singular sources in general relativity, where it encodes the well-known
+Israel junction conditions across a hypersurface on which stress-energy is concentrated [20].
+There are a few other connections between our calculations and ones that appear in the physics
+literature. The variation of the Gibbons-Hawking-York boundary term in general relativity [17, 28]
+is one example. It has many parallels to our calculations in Section 2.2, and one can undoubtedly
+find formulas like (6) in the literature after reconciling notations. We still give a full derivation
+of such formulas, not only to familiarize the reader with our notation, but also to provide careful
+derivations that refrain from discarding total derivatives (which integrate to zero on manifolds
+without boundary, but not in general) and minimize the use of local coordinate calculations where
+possible.
+4
+
+2
+Evolution of geometric quantities
+In this section, we consider an N-dimensional manifold M equipped with a smooth Riemannian
+metric g, and we study the evolution of various geometric quantities under deformations of g.
+We adopt the following notation in this section. The Levi-Civita connection associated with g
+is denoted ∇. If σ is a (p, q)-tensor field, then its covariant derivative is the (p, q + 1)-tensor field
+∇σ, and its covariant derivative in the direction of a vector field X is the (p, q)-tensor field ∇Xσ.
+Its trace Tr σ is the contraction of σ along the first two indices, using g to raise or lower indices
+as needed. We denote div σ = Tr ∇σ and ∆σ = div ∇σ. The g-inner product of two (p, q)-tensor
+fields σ and ρ is denoted ⟨σ, ρ⟩.
+The volume form associated with g is denoted ω. The Ricci tensor and the scalar curvature of g
+are denoted Ric and R, respectively. When we wish to emphasize their dependence on g, we write
+ω(g), Ric(g), R(g), etc.
+If D is an embedded submanifold of M, then we denote by ωD the induced volume form on D.
+If σ is a tensor field on M, then σ|D denotes the pullback of σ under the inclusion D ֒→ M. Later
+we will introduce some additional notation related to embedded submanifolds of codimension 1,
+like the mean curvature H and second fundamental form II; see Section 2.2.
+We denote the exterior derivative of a differential form α by dα. If α is a one-form, then α♯
+denotes the vector field obtained by raising indices with g. If f is a scalar field, then we sometimes
+interpret the one-form ∇f = df as the vector field (df)♯ without explicitly writing it.
+Later, in Section 4, we will append a subscript g to many quantities like ∇ and ⟨·, ·⟩ to emphasize
+their dependence on g.
+In that section only, an absent subscript will generally signal that the
+quantity in question is computed with respect to the Euclidean metric, which we denote by δ. We
+say more about this notational shift in Section 4.
+2.1
+Evolution of the densitized scalar curvature
+First we study the evolution of the densitized scalar curvature Rω under deformations of the metric.
+Proposition 2.1. Let g(t) be a family of smooth Riemannian metrics with time derivative ∂
+∂tg =: σ.
+We have
+∂
+∂t(Rω) = (div div Sσ)ω − ⟨G, σ⟩ω,
+where G = Ric − 1
+2Rg denotes the Einstein tensor associated with g and
+Sσ = σ − g Tr σ.
+Proof. We compute
+∂
+∂t(Rω) = ˙Rω + R ˙ω
+and invoke the well-known formulas [15, Lemma 2]
+˙R = div div σ − ∆ Tr σ − ⟨Ric, σ⟩
+and [10, Equation 2.4]
+˙ω = 1
+2(Tr σ)ω.
+Since ∆ Tr σ = div div(g Tr σ) and Tr σ = ⟨g, σ⟩, the result follows.
+5
+
+2.2
+Evolution of the mean curvature
+Next we study the evolution of the mean curvature H of a hypersurface F. We assume that the tan-
+gent bundle of F is trivial, so that there exists a smooth, g-orthonormal frame field τ1, τ2, . . . , τN−1
+on F. (If this is not the case, then one can simply fix a point p ∈ F and focus on a neighborhood of p
+on which the tangent bundle is trivial.) We let n be the unit normal to F so that n, τ1, τ2, . . . , τN−1
+forms a right-handed g-orthonormal frame (in the ambient manifold) at each point on F. If the
+metric g varies smoothly in time, then we assume that the vectors n, τ1, τ2, . . . , τN−1 also vary
+smoothly in time and remain g-orthonormal at all times.
+We use the notation
+II(X, Y ) = g(∇Xn, Y ) = −g(n, ∇XY )
+for the second fundamental form on F. Our sign convention is such that Tr II = H, and H is
+positive for a sphere with an outward normal vector. We also let ∇F and divF denote the surface
+gradient and surface divergence operators on F, which have the following meanings. For a scalar
+field v,
+∇F v = ∇v − n∇nv =
+N−1
+�
+i=1
+τi∇τiv,
+and for a one-form α,
+divF α = Tr (∇α|F) =
+N−1
+�
+i=1
+(∇τiα)(τi).
+Note that in the formula ∇F v = ∇v − n∇nv, we have regarded ∇v as a vector field rather than a
+one-form. Recall that the surface divergence operator satisfies the identity
+�
+F
+(divF α)ωF =
+�
+∂F
+α(νF )ω∂F +
+�
+F
+Hα(n)ωF ,
+(5)
+where νF is the outward unit normal to ∂F and H is the mean curvature of F.
+Proposition 2.2. Let g(t) be a family of smooth Riemannian metrics with time derivative ∂
+∂tg =: σ.
+Let F be a time-independent hypersurface with mean curvature H and induced volume form ωF.
+Then
+∂
+∂t(HωF ) = −1
+2
+��
+II, σ|F
+�
++ (div Sσ)(n) + divF (σ(n, ·)) − Hσ(n, n)
+�
+ωF,
+(6)
+where
+II(X, Y ) = II(X, Y ) − Hg(X, Y )
+is the trace-reversed second fundamental form.
+Remark 2.3. In dimension N = 2, the formula (6) simplifies considerably.
+Letting τ and n
+denote the unit tangent and unit normal to F, we have ∇τn = Hτ, −∇ττ = Hn, and II(τ, τ) =
+g(∇τn, τ) − Hg(τ, τ) = H − H = 0, so II vanishes. In addition,
+divF (σ(n, ·)) − Hσ(n, n) = ∇τ (σ(n, ·)) (τ) − Hσ(n, n)
+= ∇τ (σ(n, τ)) − σ(n, ∇ττ) − Hσ(n, n)
+= ∇τ (σ(n, τ)) .
+Thus, in two dimensions,
+∂
+∂t(HωF) = −1
+2 ((div Sσ)(n) + ∇τ (σ(n, τ))) ωF.
+6
+
+To prove Proposition 2.2, we write
+˙H = −
+N−1
+�
+i=1
+∂
+∂tg(n, ∇τiτi)
+(7)
+and use the following lemmas.
+Lemma 2.4. For any time-dependent vector fields X and Y ,
+∂
+∂t∇Y X = ∇ ˙Y X + ∇Y ˙X + 1
+2 ((∇Xσ)Y + (∇Y σ)X − (∇σ)(X, Y ))♯ ,
+where (∇σ)(X, Y ) denotes the one-form Z �→ (∇Zσ)(X, Y ), and (∇Xσ)Y denotes the one-form
+Z �→ (∇Xσ)(Y, Z).
+Proof. In coordinates,
+(∇Y X)ℓ = Y j ∂Xℓ
+∂xj + Γℓ
+ijY jXi,
+where Γℓ
+ij denote the Christoffel symbols of the second kind associated with g. Thus,
+∂
+∂t(∇Y X)ℓ = ˙Y j ∂Xℓ
+∂xj + Γℓ
+ij ˙Y jXi + Y j ∂ ˙Xℓ
+∂xj + Γℓ
+ijY j ˙Xi + ˙Γℓ
+ijY jXi
+= (∇ ˙Y X)ℓ + (∇Y ˙X)ℓ + ˙Γℓ
+ijY jXi.
+Next, we recall the following formula for the rate of change of the Christoffel symbols under a
+metric deformation [10, Equation 2.23]:
+˙Γℓ
+ij = 1
+2gℓm ((∇iσ)jm + (∇jσ)im − (∇mσ)ij) .
+It follows that
+˙Γℓ
+ijY jXi = 1
+2gℓm �
+(∇Xσ)jmY j + (∇Y σ)imXi − (∇mσ)ijY jXi�
+= 1
+2 [((∇Xσ)Y + (∇Y σ)X − (∇σ)(X, Y ))]ℓ .
+Hence,
+∂
+∂t(∇Y X)ℓ = (∇ ˙Y X)ℓ + (∇Y ˙X)ℓ + 1
+2 ((∇Xσ)Y + (∇Y σ)X − (∇σ)(X, Y ))ℓ .
+Lemma 2.5. For any time-dependent vector field X,
+∂
+∂tg(n, X) = 1
+2σ(n, n)g(n, X) + g(n, ˙X).
+Proof. Writing X = ng(n, X) + �N−1
+i=1 τig(τi, X), we compute
+∂
+∂tg(n, X) = σ(n, X) + g( ˙n, X) + g(n, ˙X)
+= σ(n, n)g(n, X) +
+N−1
+�
+i=1
+σ(n, τi)g(τi, X) + g( ˙n, n)g(n, X) +
+N−1
+�
+i=1
+g( ˙n, τi)g(τi, X) + g(n, ˙X)
+= (σ(n, n) + g( ˙n, n)) g(n, X) +
+N−1
+�
+i=1
+(σ(n, τi) + g( ˙n, τi)) g(τi, X) + g(n, ˙X).
+7
+
+For each i = 1, 2, . . . , N − 1, we have
+0 = ∂
+∂tg(n, τi) = σ(n, τi) + g( ˙n, τi) + g(n, ˙τi)
+= σ(n, τi) + g( ˙n, τi)
+since ˙τi is g-orthogonal to n. Likewise,
+0 = ∂
+∂tg(n, n) = σ(n, n) + 2g(n, ˙n),
+so the result follows.
+We are now ready to compute the time derivative of the mean curvature H. By Lemma 2.5, we
+have
+˙H = −
+N−1
+�
+i=1
+∂
+∂tg(n, ∇τiτi)
+= −
+N−1
+�
+i=1
+�1
+2σ(n, n)g(n, ∇τiτi) + g
+�
+n, ∂
+∂t∇τiτi
+��
+= 1
+2Hσ(n, n) −
+N−1
+�
+i=1
+g
+�
+n, ∂
+∂t∇τiτi
+�
+.
+(8)
+Using Lemma 2.4 and the symmetry of the second fundamental form, we can write the second term
+as
+g
+�
+n, ∂
+∂t∇τiτi
+�
+= g(n, ∇ ˙τiτi) + g(n, ∇τi ˙τi) + (∇τiσ)(n, τi) − 1
+2(∇nσ)(τi, τi)
+= 2g(n, ∇ ˙τiτi) + (∇τiσ)(n, τi) − 1
+2(∇nσ)(τi, τi).
+The first term above, when summed over i, can be simplified as follows. We write ˙τi = �N−1
+j=1 τjg(τj, ˙τi)
+and use the linearity of ∇XY in X to compute
+2
+N−1
+�
+i=1
+g(n, ∇ ˙τiτi) = 2
+N−1
+�
+i=1
+N−1
+�
+j=1
+g(n, ∇τjτi)g(τj, ˙τi)
+=
+N−1
+�
+i=1
+N−1
+�
+j=1
+g(n, ∇τjτi) (g(τj, ˙τi) + g( ˙τj, τi))
+= −
+N−1
+�
+i=1
+N−1
+�
+j=1
+g(n, ∇τjτi)σ(τj, τi)
+= ⟨II, σ|F ⟩.
+Above, we used the symmetry of the second fundamental form to pass from the first line to the
+second, and we used the identity
+0 = ∂
+∂tg(τj, τi) = σ(τj, τi) + g(τj, ˙τi) + g( ˙τj, τi)
+8
+
+to pass from the second line to the third. Inserting these results into (8), we get
+˙H = 1
+2Hσ(n, n) − ⟨II, σ|F ⟩ +
+N−1
+�
+i=1
+�1
+2(∇nσ)(τi, τi) − (∇τiσ)(n, τi)
+�
+.
+(9)
+Lemma 2.6. We have
+N−1
+�
+i=1
+�1
+2(∇nσ)(τi, τi) − (∇τiσ)(n, τi)
+�
+= 1
+2 (⟨II, σ|F⟩ − (div Sσ)(n) − divF (σ(n, ·))) .
+(10)
+Proof. The identity 0 = ∇τi (g(n, n)) = 2g(n, ∇τin) shows that ∇τin is in the span of {τj}N−1
+j=1 , so
+the first term on the right-hand side of (10) satisfies
+⟨II, σ|F ⟩ =
+N−1
+�
+i=1
+N−1
+�
+j=1
+σ(τj, τi)g(τj, ∇τin)
+=
+N−1
+�
+i=1
+σ(∇τin, τi).
+(11)
+The second term on the right-hand side of (10) can be computed as follows. Recalling that Sσ =
+σ − g Tr σ, we have
+(div Sσ)(n) = ∇n(Sσ)(n, n) +
+N−1
+�
+i=1
+∇τi(Sσ)(n, τi)
+= (∇nσ)(n, n) − ∇n(g Tr σ)(n, n) +
+N−1
+�
+i=1
+[(∇τiσ)(n, τi) − ∇τi(g Tr σ)(n, τi)]
+= (∇nσ)(n, n) − g(n, n)∇n Tr σ +
+N−1
+�
+i=1
+[(∇τiσ)(n, τi) − g(n, τi)∇τi Tr σ]
+= (∇nσ)(n, n) − ∇n Tr σ +
+N−1
+�
+i=1
+(∇τiσ)(n, τi).
+Since the trace commutes with covariant differentiation,
+∇n Tr σ = Tr ∇nσ = (∇nσ)(n, n) +
+N−1
+�
+i=1
+(∇nσ)(τi, τi).
+Thus,
+(div Sσ)(n) =
+N−1
+�
+i=1
+[(∇τiσ)(n, τi) − (∇nσ)(τi, τi)] .
+(12)
+The third term on the right-hand side of (10) is given by
+divF (σ(n, ·)) =
+N−1
+�
+i=1
+∇τi (σ(n, ·)) (τi)
+=
+N−1
+�
+i=1
+[∇τi (σ(n, τi)) − σ(n, ∇τiτi)] .
+(13)
+9
+
+Combining (11), (12), and (13), we see that
+1
+2 (⟨II, σ|F ⟩ − (div Sσ)(n) − divF (σ(n, ·)))
+= 1
+2
+N−1
+�
+i=1
+[σ(∇τin, τi) − (∇τiσ)(n, τi) + (∇nσ)(τi, τi) − ∇τi (σ(n, τi)) + σ(n, ∇τiτi)]
+= 1
+2
+N−1
+�
+i=1
+[(∇nσ)(τi, τi) − 2(∇τiσ)(n, τi)] .
+Combining Lemma 2.6 with (9), we get
+˙H = 1
+2 (−⟨II, σ|F ⟩ − (div Sσ)(n) − divF (σ(n, ·)) + Hσ(n, n)) .
+(14)
+Proposition 2.2 now follows from the identities
+∂
+∂t(HωF) = ˙HωF + H ˙ωF = ˙HωF + 1
+2H Tr (σ|F ) ωF
+and
+⟨II, σ|F ⟩ − H Tr (σ|F) = ⟨II, σ|F⟩.
+2.3
+Evolution of angles
+Next we study the evolution of angles under deformations of the metric.
+Lemma 2.7. Let g(t) be a family of smooth Riemannian metrics with time derivative
+∂
+∂tg =: σ.
+Let (¯n(t), ¯τ(t)) be a pair of g(t)-orthonormal vectors, and let (n(t), τ(t)) be another pair of g(t)-
+orthonormal vectors lying in the span of (¯n(t), ¯τ(t)). Let θ(t) be the angle for which
+τ = ¯τ cos θ + ¯n sin θ,
+n = −¯τ sin θ + ¯n cos θ.
+Assume that these vectors vary smoothly in time, and assume that n(t) (respectively, ¯n(t)) is at all
+times g(t)-orthogonal to a time-independent hypersurface F (respectively, ¯F). Then, at all times
+for which θ ∈ (0, π), we have
+∂
+∂tθ = 1
+2σ(n, τ) − 1
+2σ(¯n, ¯τ).
+(15)
+Proof. Differentiating the relation cos θ = g(¯n, n) yields
+− ˙θ sin θ = ∂
+∂t (g(¯n, n)) .
+In particular, at any time s, we can write
+− ˙θ(s) sin θ(s) = ∂
+∂t
+����
+t=s
+(g(t)(¯n(t), n(s))) + ∂
+∂t
+����
+t=s
+(g(t)(¯n(s), n(t))) − σ(s)(¯n(s), n(s)).
+10
+
+Using Lemma 2.5 and suppressing the evaluations at t = s, we get
+− ˙θ sin θ = 1
+2σ(¯n, ¯n)g(¯n, n) + 1
+2σ(n, n)g(n, ¯n) − σ(¯n, n)
+= 1
+2σ(¯n, ¯n cos θ − n) + 1
+2σ(n cos θ − ¯n, n)
+= 1
+2σ(¯n, ¯τ sin θ) + 1
+2σ(−τ sin θ, n).
+If θ ∈ (0, π) at time t = s, then we can divide by sin θ to get (15).
+3
+Distributional densitized scalar curvature
+Let T be a simplicial triangulation of a polyhedral domain Ω ⊂ RN. We use T k to denote the
+set of all k-simplices in T . We also use ˚T k to denote the subset of T k consisting of k-simplices
+that are not contained in the boundary of Ω. We call such simplices interior simplices. We call
+(N − 1)-simplices faces.
+Let g be a Regge metric on T . Recall that this means that g|T is a smooth Riemannian metric
+on each T ∈ T N, and the induced metric g|F is single-valued on each F ∈ ˚
+T N−1 (and consequently
+the induced metric is single-valued on all lower-dimensional simplices in T ).
+On each T ∈ T N, we denote by RT the scalar curvature of g|T . On an interior face F ∈ ˚
+T N−1
+that lies on the boundary of two N-simplices T + and T −, the second fundamental form on F, as
+measured by g|T +, generally differs from that measured by g|T −. We denote by �II�F the jump in
+the second fundamental form across F. More precisely,
+�II�F(X, Y ) = g|T + (∇Xn+, Y ) + g|T − (∇Xn−, Y )
+for any vectors X, Y tangent to F, where n± points outward from T ±, has unit length with respect
+to g|T ±, and is g|T ±-orthogonal to F. We adopt similar notation for the jumps in other quantities
+across F. For instance, �H�F denotes the jump in the mean curvature across F. We sometimes
+drop the subscript F when there is no danger of confusion. If F is contained in ∂Ω, then we define
+the jump in a scalar field v across F to be simply �v�F = v|F .
+On each S ∈ ˚
+T N−2, the angle defect along S is
+ΘS = 2π −
+�
+T∈T N
+T⊃S
+θST,
+where θST denotes the dihedral angle formed by the two faces of T that contain S, as measured by
+g|T . Generally this angle may vary along S. If F + and F − are the two faces of T that contain S,
+and if n± denotes the unit normal to F ± with respect to g|T pointing outward from T, then
+cos θST = − g|T (n+, n−).
+Let
+V = {v ∈ H1
+0(Ω) | ∀T ∈ T N, v|T ∈ H2(T)}.
+Note that if v ∈ V , then v admits a single-valued trace on every simplex in T of dimension ≥ N −3.
+Definition 3.1. Let g be a Regge metric. The distributional densitized scalar curvature of g is the
+linear functional (Rω)dist(g) ∈ V ′ defined by
+⟨(Rω)dist(g), v⟩V ′,V =
+�
+T∈T N
+�
+T
+RT vωT + 2
+�
+F ∈˚
+T N−1
+�
+F
+�H�FvωF + 2
+�
+S∈˚
+T N−2
+�
+S
+ΘSvωS,
+∀v ∈ V.
+(16)
+11
+
+This definition generalizes Definition 3.1 of [4], where the distributional curvature two-form (i.e.
+the Gaussian curvature times the volume form) is defined for Regge metrics in dimension N = 2.
+Note that the factors of 2 appearing in all but the first term in (16) are consistent with the fact
+that in dimension N = 2, the scalar curvature R is twice the Gaussian curvature.
+One can heuristically motivate Definition 3.1 in much the same way that one motivates its
+two-dimensional counterpart. When g is piecewise constant, Definition 3.1 recovers the classical
+notion [23] that the distributional densitized scalar curvature is a linear combination of Dirac delta
+distributions supported on (N − 2)-simplices, with weights given by angle defects. When g is not
+piecewise constant, additional terms appear which account for the nonzero (classically defined)
+curvature of g in the interior of each N-simplex T and the jump in the mean curvature across each
+interior face F. The jump in the mean curvature across F can be understood by recalling that the
+scalar curvature R at a point p ∈ F can be expressed as (two times) a sum of sectional curvatures
+of N(N − 1)/2 tangent planes that are mutually g-orthogonal at p, (N − 1)(N − 2)/2 of which
+are tangent to F at p and N − 1 of which are g-orthogonal to F at p. The sectional curvatures
+corresponding to planes tangent to F are nonsingular, owing to the tangential-tangential continuity
+of g. The remaining N − 1 sectional curvatures are singular, and by considering an N-dimensional
+region that encloses a portion of F and has small thickness in the direction that is g-orthogonal of F,
+one can use the Gauss-Bonnet theorem (along two-dimensional slices) to approximate the (volume-
+)integrated sum of these sectional curvatures by the (surface-)integrated jump in the mean curvature
+across F.
+(In this calculation, one must bear in mind that sectional curvatures and Gaussian
+curvatures are related via the Gauss-Codazzi equations.) See the discussion after Definition 3.1
+in [4], as well as [26], for more insight in dimension N = 2. See also [13] for a justification of
+Definition 3.1 in the case where g is piecewise constant and N ≥ 2.
+In the sequel, we will consistently use the letters T, F, and S to refer to simplices of dimension
+N, N − 1, and N − 2, respectively. We will therefore write �
+T , �
+F , and �
+S in place of �
+T∈T N ,
+�
+F ∈T N−1, and �
+S∈T N−2, respectively. When we wish to sum over interior simplices of a given
+dimension, we put a ring on top of the summation symbol. Thus, for example, ˚
+�
+F is shorthand
+for �
+F ∈˚
+T N−1.
+3.1
+Evolution of the distributional scalar curvature
+We are interested in how (16) changes under deformations of the metric. To this end, consider a
+one-parameter family of Regge metrics g(t) with time derivative
+σ = ∂
+∂tg.
+Our goal will be to compute
+d
+dt⟨(Rω)dist(g(t)), v⟩V ′,V
+with v ∈ V arbitrary.
+According to Propositions 2.1 and 2.2, the derivatives of the first two terms on the right-hand
+side of (16) satisfy
+d
+dt
+�
+T
+RT vωT =
+�
+T
+(div div Sσ − ⟨G, σ⟩) vωT
+and
+2 d
+dt
+�
+F
+�H�FvωF = −
+�
+F
+��
+II, σ|F
+�
++ (div Sσ)(n) + divF (σ(n, ·)) − Hσ(n, n)
+�
+vωF .
+(17)
+For the third term on the right-hand side of (16), we use the following lemma.
+12
+
+Lemma 3.2. Along any interior (N − 2)-simplex S, we have
+∂
+∂t(ΘSωS) = 1
+2
+��
+F ⊃S
+�σ(n, τ)�F + ΘS Tr(σ|S)
+�
+ωS,
+where the sum is over all (N −1)-simplices F that contain S, n is the unit normal to F with respect
+to g, and τ is the unit vector with respect to g that points into F from S and is g-orthogonal to
+both S and n. Here, our convention is that if F is shared by two N-simplices T + and T −, then
+�σ(n, τ)�F = σ+(n+, τ) + σ−(n−, τ),
+where σ± = σ|T ± and n± points outward from T ±.
+Remark 3.3. Note that n generally differs on either side of F, whereas τ does not, because g has
+single-valued tangential-tangential components along F.
+Proof. We compute
+˙ΘS = −
+�
+T⊃S
+˙θST
+and use Lemma 2.7 to differentiate each angle θST.
+The resulting expression for ˙ΘS involves
+differences between σ(n, τ) evaluated on consecutive pairs of faces F emanating from S. This sum
+can be rearranged to give
+˙ΘS = 1
+2
+�
+F ⊃S
+�σ(n, τ)�F.
+(18)
+We thus get
+∂
+∂t(ΘSωS) = ˙ΘSωS + ΘS ˙ωS
+= 1
+2
+�
+F ⊃S
+�σ(n, τ)�F ωS + 1
+2ΘS Tr (σ|S) ωS.
+It follows from the above lemma that
+2 d
+dt
+�
+S
+ΘSvωS =
+�
+S
+�
+F ⊃S
+�σ(n, τ)�F vωS +
+�
+S
+ΘS Tr(σ|S)vωS
+=
+�
+S
+�
+F ⊃S
+�σ(n, τ)�F vωS +
+�
+S
+⟨ΘSg|S, σ|S⟩ vωS.
+Collecting our results, we obtain
+d
+dt⟨(Rω)dist(g(t)), v⟩V ′,V =
+�
+T
+�
+T
+(div div Sσ)vωT
+− ˚
+�
+F
+�
+F
+�(div Sσ)(n) + divF (σ(n, ·)) − Hσ(n, n)�F vωF + ˚
+�
+S
+�
+S
+�
+F ⊃S
+�σ(n, τ)�F vωS
+(19)
+−
+�
+T
+�
+T
+⟨G, σ⟩vωT − ˚
+�
+F
+�
+F
+�
+�II�F, σ|F
+�
+vωF + ˚
+�
+S
+�
+S
+⟨ΘSg|S, σ|S⟩ vωS.
+We will now use integration by parts to rewrite the first three terms in a way that involves no
+derivatives of σ.
+13
+
+Lemma 3.4. For any v ∈ V , we have
+�
+T
+�
+T
+(div div Sσ)vωT − ˚
+�
+F
+�
+F
+�(div Sσ)(n) + divF (σ(n, ·)) − Hσ(n, n)�F vωF
++ ˚
+�
+S
+�
+S
+�
+F ⊃S
+�σ(n, τ)�FvωS =
+�
+T
+�
+T
+⟨Sσ, ∇∇v⟩ω −
+�
+F
+�
+F
+Sσ(n, n)�∇nv�ωF.
+Proof. We have
+�
+T
+�
+T
+⟨Sσ, ∇∇v⟩ω −
+�
+F
+�
+F
+Sσ(n, n)�∇nv�ωF
+(20)
+=
+�
+T
+� �
+T
+⟨Sσ, ∇∇v⟩ω −
+�
+∂T
+Sσ(n, n)∇nv ω∂T
+�
+=
+�
+T
+� �
+∂T
+Sσ(n, ∇v)ω∂T −
+�
+T
+(div Sσ)(∇v)ω −
+�
+∂T
+Sσ(n, n)∇nv ω∂T
+�
+=
+�
+T
+� �
+∂T
+Sσ(n, ∇v)ω∂T −
+�
+∂T
+(div Sσ)(n)vω∂T +
+�
+T
+(div div Sσ)vω
+−
+�
+∂T
+Sσ(n, n)∇nv ω∂T
+�
+.
+(21)
+Note that here we are regarding ∇v as a vector field rather than a one-form. On each N-simplex
+T, we can write
+�
+∂T Sσ(n, ∇v)ω∂T −
+�
+∂T Sσ(n, n)∇nv ω∂T as a sum of integrals over faces F ⊂ ∂T:
+�
+∂T
+Sσ(n, ∇v)ω∂T −
+�
+∂T
+Sσ(n, n)∇nv ω∂T =
+�
+F ⊂∂T
+�
+F
+Sσ(n, ∇v − n∇nv)ωF
+=
+�
+F ⊂∂T
+�
+F
+Sσ(n, ∇F v)ωF
+=
+�
+F ⊂∂T
+�
+F
+σ(n, ∇F v)ωF .
+In the last line above, we used the fact that ∇F v is g-orthogonal to n, so
+Sσ(n, ∇Fv) = σ(n, ∇F v) − g(n, ∇F v) Tr σ = σ(n, ∇F v).
+Each integral over F can be integrated by parts as follows. We have
+σ(n, ∇F v) = divF (σ(n, ·)v) − divF (σ(n, ·)) v,
+so the identity (5) applied to α = σ(n, ·)v implies that
+�
+F
+σ(n, ∇F v)ωF =
+�
+∂F
+σ(n, νF )vω∂F −
+�
+F
+(divF (σ(n, ·)) − Hσ(n, n)) vωF.
+Now we insert this result into (21) to get
+�
+T
+�
+T
+⟨Sσ, ∇∇v⟩ω −
+�
+F
+�
+F
+Sσ(n, n)�∇nv�ωF
+=
+�
+T
+� �
+F ⊂∂T
+�
+∂F
+σ(n, νF )vω∂F −
+�
+F ⊂∂T
+�
+F
+(divF (σ(n, ·)) − Hσ(n, n)) vωF
+−
+�
+∂T
+(div Sσ)(n)vω∂T +
+�
+T
+(div div Sσ)vω
+�
+.
+14
+
+The first term can be re-expressed as a sum over interior (N − 2)-simplices S using our notation
+from Lemma 3.2, and the next two terms can be re-expressed in terms of jumps across interior
+faces F. (Integrals over (N − 2)-simplices S ⊂ ∂Ω and (N − 1)-simplices F ⊂ ∂Ω vanish because
+v = 0 on ∂Ω.) The result is
+�
+T
+�
+T
+⟨Sσ, ∇∇v⟩ω −
+�
+F
+�
+F
+Sσ(n, n)�∇nv�ωF = ˚
+�
+S
+�
+S
+�
+F ⊃S
+�σ(n, τ)�FvωS
+− ˚
+�
+F
+�
+F
+�divF (σ(n, ·)) − Hσ(n, n) + (div Sσ)(n)� vωF +
+�
+T
+�
+T
+(div div Sσ)vω.
+Remark 3.5. Many of the above calculations are similar to the ones in [4, Proposition 4.2], except
+that here we are in dimension N rather than 2.
+We can now state the main result of this subsection.
+Theorem 3.6. Let g(t) be a family of Regge metrics with time derivative
+∂
+∂tg =: σ. For every
+v ∈ V , we have
+d
+dt⟨(Rω)dist(g(t)), v⟩V ′,V = bh(g; σ, v) − ah(g; σ, v),
+(22)
+where
+bh(g; σ, v) =
+�
+T
+�
+T
+⟨Sσ, ∇∇v⟩ω −
+�
+F
+�
+F
+Sσ(n, n)�∇nv�FωF,
+ah(g; σ, v) =
+�
+T
+�
+T
+⟨G, σ⟩vωT + ˚
+�
+F
+�
+F
+�
+�II�F, σ|F
+�
+vωF − ˚
+�
+S
+�
+S
+⟨ΘSg|S, σ|S⟩ vωS.
+Proof. Combine (19) with Lemma 3.4.
+3.2
+Distributional densitized Einstein tensor
+We now pause to make a few remarks about the bilinear forms ah(g; ·, ·) and bh(g; ·, ·) appearing
+in Theorem 3.6. These remarks will play no role in our analysis, but they help to elucidate the
+content of Theorem 3.6. The reader can safely skip ahead to Section 4 if desired.
+Numerical analysts will likely recognize the bilinear form bh(g; ·, ·) appearing in Theorem 3.6. As
+we mentioned in Section 1, it is (up to the appearance of S) a non-Euclidean, N-dimensional gener-
+alization of a bilinear form that appears in the Hellan-Herrmann-Johnson finite element method [1–
+3, 5–7, 9, 22]. It can be regarded as the integral of div div Sσ against v, where div div is interpreted
+in a distributional sense.
+The bilinear form ah(g; ·, ·) can be understood by comparing Theorem 3.6 with Proposition 2.1,
+which, when integrated against a continuous function v, states that for a family of smooth Rieman-
+nian metrics g(t) with scalar curvature R,
+d
+dt
+�
+Ω
+Rvω =
+�
+Ω
+(div div Sσ)vω −
+�
+Ω
+⟨G, σ⟩vω,
+(23)
+where σ = ∂
+∂tg and G = Ric − 1
+2Rg is the Einstein tensor associated with g. A comparison of (23)
+with (22) suggests that for a Regge metric g, the bilinear form ah(g; σ, v) should be regarded as a
+distributional counterpart of
+�
+Ω⟨G, σ⟩vω.
+15
+
+This motivates the following definition. Fix a number s > 1, and let Σ denote the space of
+square-integrable symmetric (0, 2)-tensor fields σ with the following properties: the restriction of
+σ to each T ∈ T N belongs to Hs(T), and the tangential-tangential components of σ along any
+face F ∈ ˚
+T N−1 are single-valued. Note that these conditions imply that the tangential-tangential
+components of σ along any S ∈ ˚
+T N−2 are well-defined and single-valued as well.
+Definition 3.7. Let g be a Regge metric. The distributional densitized Einstein tensor associated
+with g is the linear functional (Gω)dist(g) ∈ Σ′ defined by
+⟨(Gω)dist(g), σ⟩Σ′,Σ =
+�
+T
+�
+T
+⟨G, σ⟩ωT + ˚
+�
+F
+�
+F
+�
+�II�F, σ|F
+�
+ωF − ˚
+�
+S
+�
+S
+⟨ΘSg|S, σ|S⟩ ωS,
+∀σ ∈ Σ.
+Remark 3.8. In dimension N = 2, we have (Gω)dist(g) = 0 for any Regge metric g, because
+G vanishes within each triangle, ¯II vanishes on each edge, and the restriction of σ to each vertex
+vanishes.
+Remark 3.9. The appearance of the trace-reversed second fundamental form II in Definition 3.7
+is quite natural. The same quantity arises in studies of singular sources in general relativity, with
+the jump in II encoding the well-known Israel junction conditions across a hypersurface on which
+stress-energy is concentrated [20].
+Remark 3.10. If we define a map (div div S)dist : Σ → V ′ by
+⟨(div div S)distσ, v⟩V ′,V = bh(g; σ, v),
+∀v ∈ V,
+then, by construction, we have
+d
+dt
+����
+t=0
+⟨(Rω)dist(g + tσ), v⟩V ′,V = ⟨(div div S)distσ, v⟩V ′,V − ⟨(Gω)dist(g), vσ⟩Σ′,Σ
+for every piecewise smooth σ ∈ Σ and every smooth function v with compact support in Ω. In
+particular, suppose that Ω has no boundary (e.g., suppose that Ω is an N-dimensional cube and
+we identify its opposing faces). Then bh(g; σ, 1) = 0 and
+d
+dt
+����
+t=0
+⟨(Rω)dist(g + tσ), 1⟩V ′,V = −⟨(Gω)dist(g), σ⟩Σ′,Σ
+for every piecewise smooth σ ∈ Σ. This implies that a Regge metric g is a stationary point of
+⟨(Rω)dist(g), 1⟩Σ′,Σ if its distributional densitized Einstein tensor vanishes: (Gω)dist(g) = 0.
+The functional ⟨(Rω)dist(g), 1⟩Σ′,Σ is a counterpart of the Einstein-Hilbert functional
+�
+Ω Rω from
+general relativity, whose stationary points are solutions to the (vacuum) Einstein field equations
+G = 0. It reduces to the Regge action from Regge calculus when g is piecewise constant. That is,
+⟨(Rω)dist(g), 1⟩Σ′,Σ = 2 ˚
+�
+S
+ΘSVS,
+if g is piecewise constant,
+where VS =
+�
+S ωS denotes the volume of S. If g varies with t and remains piecewise constant for
+all t, then
+d
+dt2 ˚
+�
+S
+ΘSVS = 2 ˚
+�
+S
+˙ΘSVS + 2 ˚
+�
+S
+ΘS ˙VS,
+16
+
+and one checks that (on a domain without boundary)
+2 ˚
+�
+S
+˙ΘSVS = bh(g; σ, 1) = 0
+and
+2 ˚
+�
+S
+ΘS ˙VS = −ah(g; σ, 1) = −⟨(Gω)dist(g), σ⟩Σ′,Σ,
+where σ =
+∂
+∂tg. The fact that ˚
+�
+S ˙ΘSVS = 0 for any piecewise constant Regge metric g (on a
+domain without boundary) was proved in Regge’s classic paper [23] using very different techniques.
+Remark 3.11. If g is a Regge metric and σ = gv for some smooth function v with compact support
+in Ω, then:
+1. On each N-simplex T, we have
+⟨G, σ⟩ = ⟨G, g⟩v = (Tr G)v = −
+�N − 2
+2
+�
+Rv.
+2. On either side of each interior (N − 1)-simplex F, we have:
+�
+II, σ|F
+�
+= ⟨II, g|F ⟩ v − ⟨g|F , g|F⟩ Hv
+= Hv − (N − 1)Hv
+= −(N − 2)Hv.
+3. On each interior (N − 2)-simplex S, we have
+⟨ΘSg|S, σ|S⟩ = ΘSv Tr(g|S) = (N − 2)ΘSv.
+This shows that
+⟨(Gω)dist(g), gv⟩Σ′,Σ = −
+�N − 2
+2
+� ��
+T
+�
+T
+RT vωT + 2 ˚
+�
+F
+�
+F
+�H�FvωF + 2 ˚
+�
+S
+�
+S
+ΘSvωS
+�
+= −
+�N − 2
+2
+�
+⟨(Rω)dist(g), v⟩V ′,V
+for every smooth function v with compact support in Ω. One can interpret this as saying that the
+trace of (Gω)dist(g) is −
+� N−2
+2
+�
+(Rω)dist(g).
+Remark 3.12. If g is a piecewise constant Regge metric and σ ∈ Σ is piecewise constant, then
+⟨(Gω)dist(g), σ⟩Σ′,Σ = − ˚
+�
+S
+�
+S
+ΘS Tr(σ|S)ωS.
+If we linearize around the Euclidean metric g = δ, then we see from (18) that
+d
+dt
+����
+t=0
+⟨(Gω)dist(δ + tρ), σ⟩Σ′,Σ = − ˚
+�
+S
+�
+S
+˙ΘS Tr(σ|S)ωS
+= −1
+2
+˚
+�
+S
+�
+S
+�
+F ⊃S
+�ρ(n, τ)�F Tr(σ|S)ωS
+17
+
+for every piecewise constant ρ, σ ∈ Σ. (Note that there are no additional terms on the right-hand
+side because ΘS = 0 at t = 0.) Hence, if Ω has no boundary, then
+d2
+dt2
+����
+t=0
+⟨(Rω)dist(δ + tσ), 1⟩V ′,V = − d
+dt
+����
+t=0
+⟨(Gω)dist(δ + tσ), σ⟩Σ′,Σ
+= 1
+2
+˚
+�
+S
+�
+S
+�
+F ⊃S
+�σ(n, τ)�F Tr(σ|S)ωS
+for every piecewise constant σ ∈ Σ. This is equivalent to Christiansen’s formula [12, Theorem 2
+and Equations (25-26)] for the second variation of the Regge action around the Euclidean metric
+in dimension N = 3. (There, the Regge action is taken to be 1
+2⟨(Rω)dist(g), 1⟩V ′,V rather than
+⟨(Rω)dist(g), 1⟩V ′,V .)
+4
+Convergence
+In this section, we prove a convergence result for the distributional densitized scalar curvature in
+the norm
+∥u∥H−2(Ω) =
+sup
+v∈H2
+0(Ω),
+v̸=0
+⟨u, v⟩H−2(Ω),H2
+0(Ω)
+∥v∥H2(Ω)
+.
+(24)
+Our convergence result will be applicable to a family {gh}h>0 of Regge metrics defined on a shape-
+regular family {Th}h>0 of triangulations of Ω parametrized by h = maxT∈T N
+h hT , where hT =
+diam(T). Shape-regularity means that there exists a constant C0 independent of h such that
+max
+T∈T N
+h
+hT
+ρT
+≤ C0
+for all h > 0, where ρT denotes the inradius of T.
+Theorem 4.1. Let Ω ⊂ RN be a polyhedral domain equipped with a smooth Riemannian metric g.
+Let {gh}h>0 be a family of Regge metrics defined on a shape-regular family {Th}h>0 of triangulations
+of Ω. Assume that limh→0 ∥gh − g∥L∞(Ω) = 0 and C1 := suph>0 maxT∈T N
+h ∥gh∥W 1,∞(T) < ∞. The
+following statements hold:
+(i) If N = 2, then there exist positive constants C and h0 such that
+∥(Rω)dist(gh) − (Rω)(g)∥H−2(Ω) ≤ C
+�
+1 + max
+T
+h−1
+T ∥gh − g∥L∞(T) + max
+T
+|gh − g|W 1,∞(T)
+�
+×
+�
+∥gh − g∥2
+L2(Ω) +
+�
+T
+h2
+T |gh − g|2
+H1(T)
+�1/2
+(25)
+for all h ≤ h0. The constants C and h0 depend on ∥g∥W 1,∞(Ω), ∥g−1∥L∞(Ω), C0, and C1.
+18
+
+(ii) If N ≥ 3, assume additionally that C2 := suph>0 maxT∈T N
+h |gh|W 2,∞(T) < ∞. Then there exist
+positive constants C and h0 such that
+∥(Rω)dist(gh) − (Rω)(g)∥H−2(Ω) ≤ C
+�
+1 + max
+T
+h−2
+T ∥gh − g∥L∞(T) + max
+T
+h−1
+T |gh − g|W 1,∞(T)
+�
+×
+�
+∥gh − g∥2
+L2(Ω) +
+�
+T
+h2
+T |gh − g|2
+H1(T) +
+�
+T
+h4
+T |gh − g|2
+H2(T)
+�1/2
+(26)
+for all h ≤ h0. The constants C and h0 depend on N, ∥g∥W 1,∞(Ω), ∥g−1∥L∞(Ω), C0, C1, and
+C2.
+The above theorem leads immediately to error estimates of optimal order for piecewise poly-
+nomial interpolants of g having degree r ≥ 0, provided that either N = 2 or r ≥ 1. To make this
+statement precise, we introduce a definition. Recall that the Regge finite element space of degree
+r ≥ 0 consists of symmetric (0, 2)-tensor fields on Ω that are piecewise polynomial of degree at
+most r and possess single-valued tangential-tangential components on interior (N − 1)-simplices.
+Definition 4.2. Let Ih be a map that sends smooth symmetric (0, 2)-tensor fields on Ω to the Regge
+finite element space of degree r ≥ 0. We say that Ih is an optimal-order interpolation operator of
+degree r if there exists a number m ∈ {0, 1, . . . , N} and a constant C3 = C3(N, r, hT /ρT , t, s) such
+that for every p ∈ [1, ∞], every s ∈ (m/p, r + 1], every t ∈ [0, s], and every symmetric (0, 2)-tensor
+field g possessing W s,p(Ω)-regularity, Ihg exists (upon continuously extending Ih) and satisfies
+|Ihg − g|W t,p(T) ≤ C3hs−t
+T
+|g|W s,p(T)
+(27)
+for every T ∈ T N
+h .
+We call the number m the codimension index of Ih.
+A Regge metric gh
+is called an optimal-order interpolant of g having degree r and codimension index m if it is the
+image of a Riemannian metric g under an optimal-order interpolation operator having degree r and
+codimension index m.
+An example of an optimal-order interpolation operator is the canonical interpolation operator
+onto the degree-r Regge finite element space introduced in [21, Chapter 2]. Its degrees of freedom
+involve integrals over simplices of codimension at most N − 1, so its action on a tensor field g is
+well-defined so long as g admits traces on simplices of codimension at most N − 1, i.e. g possesses
+W s,p(Ω)-regularity with s > (N − 1)/p. Correspondingly, its codimension index is m = N − 1.
+Corollary 4.3. Let Ω, g, and {Th}h>0 be as in Theorem 4.1. Let {gh}h>0 be a family of optimal-
+order interpolants of g having degree r ≥ 0 and codimension index m. If N ≥ 3, assume that r ≥ 1.
+Then there exist positive constants C and h0 such that
+∥(Rω)dist(gh) − (Rω)(g)∥H−2(Ω) ≤ C
+��
+T
+hp(r+1)
+T
+|g|p
+W r+1,p(T)
+�1/p
+for all h ≤ h0 and all p ∈ [2, ∞] satisfying p >
+m
+r+1.
+(We interpret the right-hand side as
+C maxT hr+1
+T
+|g|W r+1,∞(T) if p = ∞.) The constants C and h0 depend on the same quantities listed
+in (i) (if N = 2) and (ii) (if N ≥ 3), as well as on Ω, r, and (if N ≥ 3) |g|W 2,∞(Ω).
+19
+
+Remark 4.4. The corollary above continues to hold if we allow slightly more general interpolants
+in Definition 4.2. For example, it holds if (27) is replaced by
+|Ihg − g|W t,p(T) ≤ C3hs−t
+T
+�
+T ′:T ′∩T̸=∅
+|g|W s,p(T ′),
+(28)
+where the sum is over all T ′ ∈ T N
+h
+that share a subsimplex with T.
+In what follows, we reuse the letter C to denote a positive constant that may change at each
+occurrence and may depend on N, ∥g∥W 1,∞(Ω), ∥g−1∥L∞(Ω), C0, and C1. Beginning in Lemma 4.8,
+we allow C to also depend on C2.
+Our strategy for proving Theorem 4.1 will be to consider an evolving metric
+�g(t) = (1 − t)g + tgh
+with time derivative
+σ = ∂
+∂t�g(t) = gh − g.
+Note that �g(t), being piecewise smooth and tangential-tangential continuous, is a Regge metric for
+all t ∈ [0, 1], and it happens to be a (globally) smooth Riemannian metric at t = 0. Since �g(0) = g
+and �g(1) = gh, Theorem 3.6 implies that
+⟨(Rω)dist(gh) − (Rω)(g), v⟩V ′,V =
+� 1
+0
+bh(�g(t); σ, v) − ah(�g(t); σ, v) dt,
+∀v ∈ V.
+Thus, we can estimate (Rω)dist(gh) − (Rω)(g) by estimating the bilinear forms bh(�g(t); ·, ·) and
+ah(�g(t); ·, ·).
+To do this, we introduce some notation. Given any Regge metric g, we let ∇g and ∇ denote
+the covariant derivatives with respect to g and δ, respectively. Similarly, we append a subscript
+g to other operators like Tr, S, and div when they are taken with respect to g, and we omit the
+subscript when they are taken with respect to δ. On the boundary of any N-simplex T, we let ng
+and n denote the outward unit normal vectors with respect to g|T and δ, respectively. These two
+vectors are related to one another in coordinates via
+ng =
+1
+�
+nT g−1n
+g−1n,
+(29)
+where we are thinking of g as a matrix and n and ng as column vectors. We write ⟨·, ·⟩g for the
+g-inner product of two tensor fields. If D is a submanifold of Ω on which the induced metric g|D
+is well-defined, and if ρ is a tensor field on D, then we denote
+∥ρ∥Lp(D,g) =
+���
+D |ρ|p
+g ωD(g)
+�1/p ,
+if 1 ≤ p < ∞,
+supD |ρ|g,
+if p = ∞,
+where ωD(g) is the induced volume form on D and |ρ|g = ⟨ρ, ρ⟩1/2
+g
+. We abbreviate ∥ · ∥Lp(D) =
+∥ · ∥Lp(D,δ) and | · | = | · |δ.
+We introduce two metric-dependent, mesh-dependent norms. For v ∈ V , we set
+∥v∥2
+2,h,g =
+�
+T
+∥∇g∇gv∥2
+L2(T,g) +
+�
+F
+h−1
+F ∥�dv(ng)�∥2
+L2(F,g) .
+20
+
+If σ is a symmetric (0, 2)-tensor field with the property that σ(ng, ng) is well-defined and single-
+valued on every F ∈ T N−1
+h
+, then we set
+∥σ∥2
+0,h,g =
+�
+T
+∥σ∥2
+L2(T,g) +
+�
+F
+hF ∥σ(ng, ng)∥2
+L2(F,g),
+where hF is the Euclidean diameter of F. Note that the image under Sg of any symmetric (0, 2)-
+tensor field possessing single-valued tangential-tangential components along faces automatically
+possesses single-valued normal-normal components along faces, because
+Sgσ(ng, ng) = σ(ng, ng) − g(ng, ng) Trg σ = − Trg (σ|F ) .
+Now we return to the setting of Theorem 4.1 and the discussion thereafter: g is a smooth
+Riemannian metric, gh is a Regge metric, �g(t) = (1 − t)g + tgh, and σ = gh − g. We assume
+throughout what follows that limh→0 ∥gh − g∥L∞(Ω) = 0 and suph>0 maxT∈T N
+h ∥gh∥W 1,∞(T) < ∞.
+These assumptions have some elementary consequences that we record here for reference (see [16]
+for a derivation). For every h sufficiently small, every t ∈ [0, 1], and every vector w with unit
+Euclidean length,
+∥�g∥L∞(Ω) + ∥�g−1∥L∞(Ω) ≤ C,
+(30)
+max
+T
+|�g|W 1,∞(T) ≤ C,
+(31)
+C−1 ≤ inf
+Ω (wT �gw) ≤ sup
+Ω
+(wT �gw) ≤ C,
+(32)
+where we are thinking of �g as a matrix and w as a column vector in the last line. Note that the
+last line implies the existence of positive lower and upper bounds on wT �g−1w as well:
+C−1 ≤ inf
+Ω (wT �g−1w) ≤ sup
+Ω
+(wT �g−1w) ≤ C.
+(33)
+In addition, the inequalities ∥�g∥L∞(Ω) ≤ C and ∥�g−1∥L∞(Ω) ≤ C imply that
+C−1∥ρ∥Lp(D,�g(t2)) ≤ ∥ρ∥Lp(D,�g(t1)) ≤ C∥ρ∥Lp(D,�g(t2))
+(34)
+and
+C−1∥ρ∥Lp(D) ≤ ∥ρ∥Lp(D,�g(t1)) ≤ C∥ρ∥Lp(D)
+(35)
+for every t1, t2 ∈ [0, 1], every admissible submanifold D, every p ∈ [1, ∞], every tensor field ρ having
+finite Lp(D)-norm, and every h sufficiently small. We select h0 > 0 so that (30-35) hold for all
+h ≤ h0, and we tacitly use these inequalities throughout our analysis.
+We will show the following near-equivalence of the norms ∥ · ∥2,h,�g and ∥ · ∥2,h,g.
+Proposition 4.5. For every v ∈ V , every h ≤ h0, and every t ∈ [0, 1],
+∥v∥2
+2,h,�g ≤ C
+�
+∥v∥2
+2,h,g +
+�
+max
+T
+h−2
+T ∥gh − g∥2
+L∞(T) + max
+T
+|gh − g|2
+W 1,∞(T)
+�
+×
+�
+T
+�
+∥dv∥2
+L2(T) + h2
+T |dv|2
+H1(T)
+� �
+.
+The proof of Proposition 4.5 relies on the following lemma.
+21
+
+Lemma 4.6. Let g1 and g2 be two symmetric positive definite matrices, and let n be a unit vector.
+Let
+ngi =
+1
+�
+nTg−1
+i
+n
+g−1
+i
+n,
+i = 1, 2.
+Then there exists a constant c depending on |g1|, |g2|, |g−1
+1 |, |g−1
+2 | such that
+|ng1 − ng2| ≤ c|g1 − g2|.
+Proof. Using the identity
+1
+�
+nT g−1
+1 n
+−
+1
+�
+nTg−1
+2 n
+=
+nT(g−1
+2
+− g−1
+1 )n
+nTg−1
+1 n
+�
+nTg−1
+2 n + nT g−1
+2 n
+�
+nT g−1
+1 n
+,
+(36)
+we can write
+ng1 − ng2 =
+nT (g−1
+2
+− g−1
+1 )n
+nT g−1
+1 n
+�
+nT g−1
+2 n + nTg−1
+2 n
+�
+nTg−1
+1 n
+g−1
+1 n +
+1
+�
+nT g−1
+2 n
+(g−1
+1
+− g−1
+2 )n.
+Since g−1
+1
+− g−1
+2
+= g−1
+1 (g2 − g1)g−1
+2 , the bound follows easily.
+Notice that in view of (29), Lemma 4.6 implies that
+∥n�g − ng∥L∞(F ) ≤ C∥�g − g∥L∞(F )
+(37)
+on either side of any face F.
+Now we are ready to begin proving Proposition 4.5. Consider the term �
+F h−1
+F
+���dv(n�g)�
+��2
+L2(F,�g)
+that appears in the definition of ∥v∥2
+2,h,�g. Notice that
+dv(n�g) = dv(ng) + dv(n�g − ng),
+and we can use the bound (37) to estimate
+∥dv(n�g − ng)∥L2(F,�g) ≤ C∥dv(n�g − ng)∥L2(F )
+≤ C∥dv∥L2(F )∥n�g − ng∥L∞(F )
+≤ C∥dv∥L2(F )∥�g − g∥L∞(F )
+≤ C∥dv∥L2(F )∥gh − g∥L∞(F )
+on either side of F. Using the trace inequality
+∥dv∥2
+L2(F ) ≤ C
+�
+h−1
+T ∥dv∥2
+L2(T) + hT |dv|2
+H1(T)
+�
+,
+F ⊂ T ∈ T N
+h ,
+(38)
+it follows that
+�
+F
+h−1
+F ∥�dv(n�g)�∥2
+L2(F,�g)
+≤ C
+��
+F
+h−1
+F ∥�dv(ng)�∥2
+L2(F,g) +
+�
+T
+h−1
+T
+�
+h−1
+T ∥dv∥2
+L2(T) + hT |dv|2
+H1(T)
+�
+∥gh − g∥2
+L∞(T)
+�
+= C
+��
+F
+h−1
+F ∥�dv(ng)�∥2
+L2(F,g) +
+�
+T
+�
+h−2
+T ∥gh − g∥2
+L∞(T)∥dv∥2
+L2(T) + ∥gh − g∥2
+L∞(T)|dv|2
+H1(T)
+��
+,
+22
+
+where we have used (34), (38), and the bound hT ≤ ChF, which follows from the shape-regularity
+of Th.
+Next, consider the term �
+T ∥∇�g∇�gv∥2
+L2(T,�g) that appears in the definition of ∥v∥2
+2,h,�g. Notice
+that
+�
+∇�g∇�gv
+�
+ij = (∇g∇gv)ij + (Γk
+ij − �Γk
+ij) ∂v
+∂xk ,
+where Γk
+ij and �Γk
+ij are the Christoffel symbols of the second kind associated with g and �g, respec-
+tively. We have
+∥Γk
+ij − �Γk
+ij∥L∞(T) ≤ C∥�g − g∥W 1,∞(T) ≤ C∥gh − g∥W 1,∞(T),
+so
+∥∇�g∇�gv∥L2(T,�g) ≤ C∥∇�g∇�gv∥L2(T)
+≤ C
+�
+∥∇g∇gv∥L2(T) + ∥gh − g∥W 1,∞(T)∥dv∥L2(T)
+�
+≤ C
+�
+∥∇g∇gv∥L2(T,g) + ∥gh − g∥W 1,∞(T)∥dv∥L2(T)
+�
+.
+It follows that
+∥v∥2
+2,h,�g ≤ C
+�
+∥v∥2
+2,h,g +
+�
+max
+T
+h−2
+T ∥gh − g∥2
+L∞(T) + max
+T
+|gh − g|2
+W 1,∞(T)
+�
+×
+�
+T
+�
+∥dv∥2
+L2(T) + h2
+T |dv|2
+H1(T)
+� �
+.
+This completes the proof of Proposition 4.5.
+Our next step will be to estimate the bilinear form bh(�g; ·, ·).
+Proposition 4.7. For every h ≤ h0, every t ∈ [0, 1], and every v ∈ H2
+0(Ω), we have (with
+σ = gh − g)
+|bh(�g; σ, v)| ≤ C
+�
+∥gh − g∥2
+L2(Ω) +
+�
+T
+h2
+T |gh − g|2
+H1(T)
+�1/2
+×
+�
+1 + max
+T
+h−1
+T ∥gh − g∥L∞(T) + max
+T
+|gh − g|W 1,∞(T)
+�
+∥v∥H2(Ω).
+Proof. In view of the definitions of ∥ · ∥0,h,�g and ∥ · ∥2,h,�g, we have
+|bh(�g; σ, v)| ≤ ∥S�gσ∥0,h,�g∥v∥2,h,�g.
+(39)
+Recalling that
+∥S�gσ∥2
+0,h,�g =
+�
+T
+∥S�gσ∥2
+L2(T,�g) +
+�
+F
+hF ∥S�gσ(n�g, n�g)∥2
+L2(F,�g),
+we compute
+⟨S�gσ, S�gσ⟩�g =
+�
+σ − �g⟨�g, σ⟩�g, σ − �g⟨�g, σ⟩�g
+�
+�g
+= ⟨σ, σ⟩�g − 2⟨�g, σ⟩2
+�g + ⟨�g, �g⟩�g⟨�g, σ⟩2
+�g
+= ⟨σ, σ⟩�g + (N − 2)⟨�g, σ⟩2
+�g,
+23
+
+which leads to the bound
+∥S�gσ∥L2(T,�g) ≤ C∥σ∥L2(T,�g) ≤ C∥σ∥L2(T).
+Also, by the trace inequality,
+∥S�gσ(n�g, n�g)∥2
+L2(∂T,�g) ≤ C∥S�gσ∥2
+L2(∂T,�g)
+≤ C∥σ∥2
+L2(∂T)
+≤ C
+�
+h−1
+T ∥σ∥2
+L2(T) + hT |σ|2
+H1(T)
+�
+.
+(Here we are measuring the L2(∂T, �g)-norm of the full tensor S�gσ rather than its restriction to the
+tangent bundle of ∂T.) Thus,
+∥S�gσ∥2
+0,h,�g ≤ C
+�
+∥σ∥2
+L2(Ω) +
+�
+T
+h2
+T |σ|2
+H1(T)
+�
+= C
+�
+∥gh − g∥2
+L2(Ω) +
+�
+T
+h2
+T |gh − g|2
+H1(T)
+�
+.
+(40)
+Consider now the term ∥v∥2,h,�g in (39). Proposition 4.5 implies that
+∥v∥2,h,�g ≤ C
+�
+∥v∥2,h,g +
+�
+max
+T
+h−1
+T ∥gh − g∥L∞(T) + max
+T
+|gh − g|W 1,∞(T)
+�
+∥v∥H2(Ω)
+�
+since v ∈ H2
+0(Ω).
+Furthermore, since g is smooth and v ∈ H2
+0(Ω), we have �dv(ng)� = 0 on
+every interior face F and �dv(ng)� = dv(ng) = 0 on every face F ⊂ ∂Ω.
+Thus, ∥v∥2
+2,h,g =
+�
+T ∥∇g∇gv∥2
+L2(T,g) = ∥∇g∇gv∥2
+L2(Ω,g). Since
+(∇g∇gv)ij = (∇∇v)ij − Γk
+ij
+∂v
+∂xk ,
+we see that
+∥v∥2,h,g = ∥∇g∇gv∥L2(Ω) ≤ C(|v|H2(Ω) + |v|H1(Ω)) ≤ C∥v∥H2(Ω).
+Thus,
+∥v∥2,h,�g ≤ C
+�
+1 + max
+T
+h−1
+T ∥gh − g∥L∞(T) + max
+T
+|gh − g|W 1,∞(T)
+�
+∥v∥H2(Ω).
+(41)
+Combining (39), (40), and (41) completes the proof.
+At this point, we have finished proving part (i) of Theorem 4.1. Indeed, in dimension N = 2,
+ah vanishes, so we can write
+��⟨(Rω)dist(gh) − (Rω)(g), v⟩V ′,V
+�� ≤
+� 1
+0
+|bh(�g(t); σ, v)| dt
+and apply Proposition 4.7 to deduce (25).
+To prove part (ii) of Theorem 4.1, we suppose that N ≥ 3 and that suph>0 maxT∈T N
+h |gh|W 2,∞(T) <
+∞, and we proceed as follows. Recall that
+ah(�g; σ, v) =
+�
+T
+�
+T
+⟨G(�g), σ⟩�gvωT (�g)+ ˚
+�
+F
+�
+F
+�
+�II(�g)�F, σ|F
+�
+�g vωF(�g)− ˚
+�
+S
+�
+S
+⟨ΘS(�g)�g|S, σ|S⟩�gvωS(�g),
+(42)
+24
+
+where have made all dependencies on the metric explicit in the notation. We will bound each of
+the three terms above, beginning with the first. Throughout what follows, we continue to denote
+σ = gh − g, and we let v be an arbitrary member of V .
+Lemma 4.8. We have
+�����
+�
+T
+�
+T
+⟨G(�g), σ⟩�g vωT (�g)
+����� ≤ C∥gh − g∥L2(Ω)∥v∥L2(Ω).
+Proof. Since we are now assuming that suph>0 maxT∈T N
+h ∥gh∥W 2,∞(T) < ∞, the Einstein tensor
+associated with �g satisfies
+∥G(�g)∥L∞(T) ≤ C
+for every h ≤ h0, every t ∈ [0, 1], and every T ∈ T N
+h . It follows that
+����
+�
+T
+⟨G(�g), σ⟩�g vωT (�g)
+���� ≤ ∥G(�g)∥L∞(T,�g)∥σ∥L2(T,�g)∥v∥L2(T,�g)
+≤ C∥G(�g)∥L∞(T)∥σ∥L2(T)∥v∥L2(T)
+≤ C∥σ∥L2(T)∥v∥L2(T)
+= C∥gh − g∥L2(T)∥v∥L2(T).
+Summing over all T ∈ T N
+h
+completes the proof.
+Lemma 4.9. We have
+�����
+˚
+�
+F
+�
+F
+�
+�II(�g)�F, σ|F
+�
+�g vωF(�g)
+����� ≤ C max
+T
+�
+h−1
+T ∥gh − g∥W 1,∞(T)
+�
+×
+��
+T
+∥gh − g∥2
+L2(T) + h2
+T |gh − g|2
+H1(T)
+�1/2 ��
+T
+∥v∥2
+L2(T) + h2
+T |v|2
+H1(T)
+�1/2
+.
+Proof. Consider an interior (N − 1)-simplex F. By applying a Euclidean rotation and translation
+to the coordinates, we may assume without loss of generality that F lies in the plane xN = 0. In
+these coordinates, the second fundamental form associated with �g is given by
+IIij(�g) = −�g(n�g, ∇�g,eiej)
+= −�g(n�g, �Γk
+ijek)
+= −nℓ
+�g�gℓk�Γk
+ij,
+i, j = 1, 2, . . . , N − 1,
+where e1, e2, . . . , eN are the Euclidean coordinate basis vectors. Since n�g = �g−1n/
+�
+nT �g−1n and n
+points in the xN direction, we get
+IIij(�g) = −
+1
+�
+nT �g−1n
+�ΓN
+ij .
+The jump in this quantity across F can be computed using the identity �ab� = �a�{b} + {a}�b�,
+where {·} denotes the average across F, giving
+−�IIij(�g)� =
+�
+1
+�
+nT �g−1n
+� �
+�ΓN
+ij
+�
++
+�
+1
+�
+nT �g−1n
+� �
+�ΓN
+ij
+�
+.
+25
+
+In view of (36), we have
+�����
+�
+1
+�
+nT �g−1n
+������
+L∞(F )
+≤ C ∥��g�∥L∞(F )
+≤ C ∥�gh − g�∥L∞(F )
+≤ C
+�
+∥gh − g∥L∞(T1) + ∥gh − g∥L∞(T2)
+�
+,
+where T1 and T2 are the two N-simplices that share the face F.
+Here, we used the fact that
+�g = g + t(gh − g) and g is smooth. Similarly, we have
+���
+�
+�ΓN
+ij
+����
+L∞(F ) ≤ C∥��g�∥W 1,∞(F )
+≤ C∥�gh − g�∥W 1,∞(F )
+≤ C
+�
+∥gh − g∥W 1,∞(T1) + ∥gh − g∥W 1,∞(T2)
+�
+.
+(43)
+Thus,
+∥�II(�g)�∥L∞(F ) ≤ C
+�
+∥gh − g∥W 1,∞(T1) + ∥gh − g∥W 1,∞(T2)
+�
+.
+From this it follows easily that the same bound holds, possibly with a larger constant C, for the
+trace-reversed tensor II(�g) = II(�g) − H(�g)�g:
+∥�II(�g)�∥L∞(F ) ≤ C
+�
+∥gh − g∥W 1,∞(T1) + ∥gh − g∥W 1,∞(T2)
+�
+.
+It follows that
+����
+�
+F
+�
+�II(�g)�F , σ|F
+�
+�g vωF (�g)
+����
+≤ ∥�II(�g)�∥L∞(F,�g)∥σ|F ∥L2(F,�g)∥v∥L2(F,�g)
+≤ C∥�II(�g)�∥L∞(F )∥σ|F ∥L2(F )∥v∥L2(F )
+≤ C
+� 2
+�
+i=1
+∥gh − g∥W 1,∞(Ti)
+� �
+h−1
+T1 ∥σ∥2
+L2(T1) + hT1|σ|2
+H1(T1)
+�1/2 �
+h−1
+T1 ∥v∥2
+L2(T1) + hT1|v|2
+H1(T1)
+�1/2
+.
+By the shape-regularity of Th, we have C−1 ≤ hT1/hT2 ≤ C for some constant C independent of h
+and F, so
+�����
+˚
+�
+F
+�
+F
+�
+�II(�g)�F, σ|F
+�
+�g vωF(�g)
+����� ≤ C max
+T
+�
+h−1
+T ∥gh − g∥W 1,∞(T)
+�
+×
+��
+T
+∥gh − g∥2
+L2(T) + h2
+T |gh − g|2
+H1(T)
+�1/2 ��
+T
+∥v∥2
+L2(T) + h2
+T |v|2
+H1(T)
+�1/2
+.
+Remark 4.10. If gh is piecewise constant, then in (43) we have the sharper bound
+∥�gh − g�∥W 1,∞(F ) = ∥�gh − g�∥L∞(F ) ≤ C
+�
+∥gh − g∥L∞(T1) + ∥gh − g∥L∞(T2)
+�
+26
+
+because ∂gh
+∂xi = 0 and
+∂g
+∂xi is continuous for each i. This implies that for piecewise constant gh, we
+can replace ∥gh − g∥W 1,∞(T) by ∥gh − g∥L∞(T) in Lemma 4.9, yielding
+�����
+˚
+�
+F
+�
+F
+�
+�II(�g)�F, σ|F
+�
+�g vωF(�g)
+����� ≤ C max
+T
+�
+h−1
+T ∥gh − g∥L∞(T)
+�
+×
+��
+T
+∥gh − g∥2
+L2(T) + h2
+T |gh − g|2
+H1(T)
+�1/2 ��
+T
+∥v∥2
+L2(T) + h2
+T |v|2
+H1(T)
+�1/2
+.
+Now we turn our attention toward the third integral in (42). In preparation for this, we will
+first use the shape-regularity assumption to show that the dihedral angles of every N-simplex in
+Th (measured in the Euclidean metric) are uniformly bounded above and below.
+Lemma 4.11. There exist constants θmin, θmax ∈ (0, π) such that for every h > 0 and every
+T ∈ T N
+h , the dihedral angles in T (measured in the Euclidean metric) all lie between θmin and θmax.
+Proof. This fact is proved in dimension N = 3 in [18, Lemma 3.6].
+We generalize their proof
+to dimension N ≥ 3 as follows. Given N + 1 points x1, x2, . . . , xN+1 in general position in RN,
+let T = [x1, x2, . . . , xN+1] denote the N-simplex with vertices x1, x2, . . . , xN+1.
+Consider two
+faces F1 = [x1, x3, x4, . . . , xN+1] and F2 = [x2, x3, x4, . . . , xN+1] that intersect along the (N − 2)-
+dimensional subsimplex S = [x3, x4, . . . , xN+1]. Throughout what follows, we work in the Euclidean
+metric. Let A be the orthogonal projection of x1 onto the (N−1)-dimensional hyperplane containing
+F2, and let B be the orthogonal projection of x1 onto the (N −2)-dimensional hyperplane containing
+S. Observe that both [x1, A] and [x1, B] are orthogonal to S, since S ⊂ F2. Thus, the triangle
+[x1, A, B] is orthogonal to S. This triangle is a right triangle with hypotenuse [x1, B], so the dihedral
+angle θST along S satisfies
+sin θST = |[x1, A]|
+|[x1, B]|,
+where | · | denotes the Euclidean volume (i.e. length in this case). Obviously, |[x1, B]| is bounded
+above by hT , the diameter of T. In addition, |[x1, A]| is bounded from below by 2 times ρT , the
+inradius of T. To see why, we generalize the argument in [18, Proposition 2.3], bearing in mind
+that our definition of ρT differs from theirs by a factor of 2. Consider the inscribed (N − 1)-sphere
+in T, whose center C lies at a distance ρT from F2. Let D be the point where this inscribed sphere
+touches F2, and let E be the point diametrically opposite to D on this sphere. The line segment
+[D, E] is orthogonal to F2, so the volume of the N-simplex T ′ = [E, x2, x3, x4, . . . , xN+1] satisfies
+|T ′| = 1
+N |[D, E]||F2| = 2ρT
+N |F2|.
+Since T ′ ⊂ T, we have
+|T ′| ≤ |T| = 1
+N |[x1, A]||F2|,
+so
+2ρT ≤ |[x1, A]|.
+Thus,
+sin θST ≥ 2ρT
+hT
+.
+The result follows from this bound and the shape-regularity of Th.
+27
+
+Next we show that Lemma 4.11 remains valid when one measures angles with g rather than the
+Euclidean metric δ.
+Lemma 4.12. Upon reducing the value of h0 if necessary, there exist constants θmin,g, θmax,g ∈ (0, π)
+such that for every h ≤ h0, every T ∈ T N
+h , every (N − 2)-simplex S ⊂ ∂T, and every point p ∈ S,
+the dihedral angle in T at p (measured by g) lies between θmin,g and θmax,g.
+Proof. If there were no such lower bound θmin,g > 0, then there would exist a sequence of N-
+simplices T1 ∈ Th1, T2 ∈ Th2, . . . with faces F (1)
+1
+, F (2)
+1
+⊂ T1, F (1)
+2 , F (2)
+2
+⊂ T2, . . . and points
+p1 ∈ F (1)
+1
+∩ F (2)
+1 , p2 ∈ F (1)
+2
+∩ F (2)
+2
+, . . . such that
+∠ g|Ti(pi)(F (1)
+i
+, F (2)
+i
+) → 0
+as i → ∞, where ∠g(X, Y ) denotes the angle between X and Y as measured by g. Using the
+compactness of the Grassmannian, this implies that, after extracting a subsequence which we do
+not relabel,
+∠δ(F (1)
+i
+, F (2)
+i
+) → 0,
+where ∠δ(X, Y ) denotes the angle between X and Y as measured by the Euclidean metric δ. This
+contradicts the assumed positive lower bound on the Euclidean dihedral angles. The existence of
+an upper bound θmax,g < π is proved similarly.
+Now we are ready to estimate the third integral in (42).
+Lemma 4.13. We have
+�����
+˚
+�
+S
+�
+S
+⟨ΘS(�g) �g|S , σ|S⟩�g vωS(�g)
+�����
+≤ C
+�
+max
+T
+h−2
+T ∥gh − g∥L∞(T)
+� ��
+T
+∥gh − g∥2
+L2(T) + h2
+T |gh − g|2
+H1(T) + h4
+T |gh − g|2
+H2(T)
+�1/2
+×
+��
+T
+∥v∥2
+L2(T) + h2
+T |v|2
+H1(T) + h4
+T |v|2
+H2(T)
+�1/2
+.
+Proof. Fix an interior (N − 2)-simplex S and an N-simplex T containing S. At any point p along
+S, we have
+cos θST(g) − cos θST (�g) = �g(n(1)
+�g , n(2)
+�g ) − g(n(1)
+g , n(2)
+g )
+= �g(n(1)
+�g
+− n(1)
+g , n(2)
+�g
+− n(2)
+g ) + �g(n(1)
+�g
+− n(1)
+g , n(2)
+g ) + �g(n(1)
+g , n(2)
+�g
+− n(2)
+g )
++ �g(n(1)
+g , n(2)
+g ) − g(n(1)
+g , n(2)
+g ),
+where n(1)
+g
+and n(2)
+g
+are suitably oriented unit normal vectors (with respect to g|T ) to the two faces
+of T containing S, and similarly for n(1)
+�g
+and n(2)
+�g . Using Lemma 4.6, we see that at the point p,
+| cos θST(�g) − cos θST(g)| ≤ C|�g − g| ≤ C|gh − g|
+for all h sufficiently small.
+Since there are constants θmin,g, θmax,g ∈ (0, π) such that θmin,g ≤
+θST(g) ≤ θmax,g, we get
+|θST (�g) − θST(g)| ≤ C|gh − g| ≤ C∥gh − g∥L∞(T).
+28
+
+Summing over T ⊃ S and noting that �
+T⊃S θST (g) = 2π, we get
+|ΘS(�g)| = |ΘS(�g) − ΘS(g)| ≤
+�
+T⊃S
+|θST(�g) − θST(g)| ≤ C
+�
+T⊃S
+∥gh − g∥L∞(T).
+(44)
+Now we are almost ready to estimate the integral
+�
+S ⟨ΘS(�g) �g|S , σ|S⟩�g vωS(�g). We first note that
+∥v∥2
+L2(S) ≤ C
+�
+h−2
+T ∥v∥2
+L2(T) + |v|2
+H1(T) + h2
+T |v|2
+H2(T)
+�
+,
+which can be proved using a codimension-2 trace inequality and a scaling argument, or by applying
+the codimension-1 trace inequality (38) twice (to v rather than dv).
+If T1, T2, . . . , Tm are the
+N-simplices that share the (N − 2)-simplex S, then we have
+����
+�
+S
+⟨ΘS(�g) �g|S , σ|S⟩�g vωS(�g)
+����
+≤ C∥ΘS(�g)∥L∞(S,�g)∥σ|S∥L2(S,�g)∥v∥L2(S,�g)
+≤ C∥ΘS(�g)∥L∞(S)∥σ|S∥L2(S)∥v∥L2(S)
+≤ C
+� m
+�
+i=1
+∥gh − g∥L∞(Ti)
+� �
+h−2
+T1 ∥σ∥2
+L2(T1) + |σ|2
+H1(T1) + h2
+T1|σ|2
+H2(T1)
+�1/2
+×
+�
+h−2
+T1 ∥v∥2
+L2(T1) + |v|2
+H1(T1) + h2
+T1|v|2
+H2(T1)
+�1/2
+.
+The proof is completed by summing over all interior (N − 2)-simplices S and substituting σ =
+gh − g.
+Collecting our results, we can state a bound on the bilinear form ah(�g; ·, ·).
+Proposition 4.14. For every h ≤ h0, every t ∈ [0, 1], and every v ∈ V , we have (with σ = gh −g),
+|ah(�g; σ, v)| ≤ C
+�
+1 + max
+T
+h−2
+T ∥gh − g∥L∞(T) + max
+T
+h−1
+T |gh − g|W 1,∞(T)
+�
+×
+��
+T
+∥gh − g∥2
+L2(T) + h2
+T |gh − g|2
+H1(T) + h4
+T |gh − g|2
+H2(T)
+�1/2
+×
+��
+T
+∥v∥2
+L2(T) + h2
+T |v|2
+H1(T) + h4
+T |v|2
+H2(T)
+�1/2
+.
+Proof. Combine Lemmas 4.8, 4.9, and 4.13.
+Upon combining Proposition 4.7 with Proposition 4.14, we see that
+∥(Rω)dist(gh) − (Rω)(g)∥H−2(Ω) ≤ C
+�
+1 + max
+T
+h−2
+T ∥gh − g∥L∞(T) + max
+T
+h−1
+T |gh − g|W 1,∞(T)
+�
+×
+�
+∥gh − g∥2
+L2(Ω) +
+�
+T
+h2
+T |gh − g|2
+H1(T) +
+�
+T
+h4
+T |gh − g|2
+H2(T)
+�1/2
+.
+29
+
+This completes the proof of Theorem 4.1. Corollary 4.3 then follows from (27) and the bounds
+∥gh − g∥L2(Ω) ≤ |Ω|1/2−1/p∥gh − g∥Lp(Ω),
+��
+T
+h2
+T |gh − g|2
+H1(T)
+�1/2
+≤ |Ω|1/2−1/p
+��
+T
+hp
+T |gh − g|p
+W 1,p(T)
+�1/p
+,
+��
+T
+h4
+T |gh − g|2
+H2(T)
+�1/2
+≤ |Ω|1/2−1/p
+��
+T
+h2p
+T |gh − g|p
+W 2,p(T)
+�1/p
+,
+which hold for all p ∈ [2, ∞] (with the obvious modifications for p = ∞).
+Remark 4.15. Notice that the analysis above yields
+|bh(�g; σ, v)| = O(hr+1),
+(by Proposition 4.7),
+(45)
+�����
+�
+T
+�
+T
+⟨G(�g), σ⟩�g vωT (�g)
+����� = O(hr+1),
+(by Lemma 4.8),
+(46)
+�����
+˚
+�
+F
+�
+F
+�
+�II(�g)�F , σ|F
+�
+�g vωF (�g)
+����� =
+�
+O(h),
+if r = 0,
+O(h2r),
+if r ≥ 1,
+(by Remark 4.10),
+(by Lemma 4.9),
+(47)
+�����
+˚
+�
+S
+�
+S
+⟨ΘS(�g) �g|S , σ|S⟩�g vωS(�g)
+����� = O(h2r),
+(by Lemma 4.13)
+(48)
+for any optimal-order interpolant gh of g having degree r ≥ 0. Bearing in mind that (46-48) vanish
+when N = 2, we see that the above estimates lead to an optimal error estimate ∥(Rω)dist(gh) −
+(Rω)(g)∥H−2(Ω) = O(hr+1) in all cases except when N ≥ 3 and r = 0, where we obtain ∥(Rω)dist(gh)−
+(Rω)(g)∥H−2(Ω) = O(1) because of (48). Numerical experiments suggest that these analytical re-
+sults are sharp for a general optimal-order interpolant, whereas for the canonical interpolant the
+estimate (48) improves to O(h2(r+1)), yielding ∥(Rω)dist(gh) − (Rω)(g)∥H−2(Ω) = O(h) when r = 0;
+cf. Figure 2.
+5
+Numerical examples
+In this section we present numerical experiments in dimension N = 2, 3 to illustrate the predicted
+convergence rates. The examples were performed in the open source finite element library NGSolve1
+[24, 25], where the Regge finite elements are available for arbitrary polynomial order. We construct
+an optimal-order interpolant gh of a given metric tensor g as follows. On each element T, the local
+L2 best-approximation ¯gh|T of g|T is computed. Then the tangential-tangential degrees of freedom
+shared by two or more neighboring elements are averaged to obtain a globally tangential-tangential
+continuous interpolant gh.
+We verify in Appendix A that this interpolant is an optimal-order
+interpolant in the sense of Remark 4.4 on shape-regular, quasi-uniform triangulations.
+To compute the H−2(Ω)-norm of the error f := (Rω)dist(gh) − (Rω)(g) we make use of the fact
+that ∥f∥H−2(Ω) is equivalent to ∥u∥H2(Ω), where u ∈ H2
+0(Ω) solves the biharmonic equation ∆2u = f.
+This equation will be solved numerically using the (Euclidean) Hellan–Herrmann–Johnson method.
+To prevent the discretization error from spoiling the real error, we use for uh two polynomial orders
+more than for gh.
+1www.ngsolve.org
+30
+
+We consider in dimension N = 2 the numerical example proposed in [16], where on the square
+Ω = (−1, 1)2 the smooth Riemannian metric tensor
+g(x, y) :=
+�
+1 + (∂f
+∂x)2
+∂f
+∂x
+∂f
+∂y
+∂f
+∂x
+∂f
+∂y
+1 + (∂f
+∂y )2
+�
+with f(x, y) := 1
+2(x2 + y2) − 1
+12(x4 + y4) is defined. This metric corresponds to the surface induced
+by the embedding
+�
+x, y
+�
+�→
+�
+x, y, f(x, y)
+�
+, and its exact scalar curvature is given by
+R(g)(x, y) =
+162(1 − x2)(1 − y2)
+(9 + x2(x2 − 3)2 + y2(y2 − 3)2)2 .
+For a three-dimensional example we consider the cube Ω = (−1, 1)3 and the Riemannian metric
+tensor induced by the embedding
+�
+x, y, z
+�
+�→
+�
+x, y, z, f(x, y, z)
+�
+, where f(x, y, z) := 1
+2(x2 + y2 +
+z2) − 1
+12(x4 + y4 + z4). The scalar curvature is
+R(g)(x, y, z) = 18
+�
+(1 − x2)(1 − y2)(9 + q(z)) + (1 − y2)(1 − z2)(9 + q(x)) + (1 − z2)(1 − x2)(9 + q(y))
+�
+(9 + q(x) + q(y) + q(z))2
+,
+where q(x) = x2(x2 − 3)2.
+We start with a structured mesh consisting of 2·22k triangles and 6·23k tetrahedra, respectively,
+in two and three dimensions with ˜h = maxT hT =
+√
+N 21−k (and minimal edge length 21−k)
+for k = 0, 1, . . . . To avoid possible superconvergence due to mesh symmetries, we perturb each
+component of the inner mesh vertices by a random number drawn from a uniform distribution in
+the range [−˜h 2−(2N+1)/2, ˜h 2−(2N+1)/2]. As depicted in Figure 1 (left) and listed in Table 1, linear
+convergence is observed when N = 2 and gh has polynomial degree r = 0. This is consistent with
+Theorem 4.1(i). For r = 1 and r = 2, higher convergence rates are obtained as expected.
+In the three-dimensional case, the same convergence rates as for N = 2 are obtained, cf. Figure 1
+(right) and Table 2. This indicates that Theorem 4.1(ii) is sharp for r ≥ 1. For r = 0 we observe
+numerically linear convergence, which is better than predicted by Theorem 4.1(ii). However, further
+investigation suggests that the observed linear convergence for r = 0 is pre-asymptotic. Indeed, to
+test if (48) is sharp, we compute the H−2(Ω)-norm of the linear functional
+v �→
+� 1
+0
+˚
+�
+S
+�
+S
+⟨ΘS(�g(t)) �g(t)|S , σ|S⟩�g(t) vωS(�g(t)) dt,
+(49)
+where we approximate the parameter integral by a Gauss quadrature of order seven. As depicted
+in Figure 2, the norm of this functional for the optimal-order interpolant gh with r = 0 stagnates at
+about 4·10−4, which is below the overall error of 4.296·10−3 for the finest grid; cf. Table 2. There-
+fore, the lack of convergence predicted by Theorem 4.1(ii) is not yet visible in Figure 1. For r = 1, 2
+the proven rate of O(h2r) for (49) (see (48)) is clearly obtained. Interestingly, using the canonical
+interpolant appears to increase the convergence rate of (49) to O(h2(r+1)) (i.e. an increase of two
+orders), as observed in Figure 2. Thus, it appears that the canonical interpolant achieves conver-
+gence in the lowest-order case. We intend to study this superconvergence phenomenon exhibited
+by the canonical interpolant in future work.
+Acknowledgments
+We thank Yasha Berchenko-Kogan for many helpful discussions, especially about the mean curva-
+ture term in Definition 3.1. We also thank Snorre Christiansen for pointing out the link with the
+31
+
+101
+102
+103
+104
+105
+106
+10−8
+10−6
+10−4
+10−2
+100
+ndof
+error
+r = 0
+r = 1
+r = 2
+O(h)
+O(h2)
+O(h3)
+101
+102
+103
+104
+105
+106
+107
+108
+10−6
+10−5
+10−4
+10−3
+10−2
+10−1
+100
+ndof
+error
+r = 0
+r = 1
+r = 2
+O(h)
+O(h2)
+O(h3)
+Figure 1: Convergence of the distributional scalar curvature in the H−2(Ω)-norm for N = 2 (left)
+and N = 3 (right) with respect to the number of degrees of freedom (ndof) of gh for r = 0, 1, 2.
+101
+102
+103
+104
+105
+106
+107
+10−12
+10−10
+10−8
+10−6
+10−4
+10−2
+ndof
+H−2(Ω)-norm of (49)
+r = 0
+r = 1
+r = 2
+r = 0 c.i.
+r = 1 c.i.
+r = 2 c.i.
+O(h2)
+O(h4)
+O(h6)
+Figure 2: Convergence of (49) in the H−2(Ω)-norm with respect to number of degrees of freedom
+(ndof) for an optimal-order interpolant and the canonical interpolant (c.i.)
+for r = 0, 1, 2 in
+dimension N = 3.
+r = 0
+r = 1
+r = 2
+h
+Error
+Order
+Error
+Order
+Error
+Order
+2.828 · 10−0
+1.534 · 10−0
+8.584 · 10−1
+4.609 · 10−1
+2.417 · 10−1
+1.251 · 10−1
+6.260 · 10−2
+3.198 · 10−2
+2.237 · 10−1
+1.945 · 10−1
+0.23
+6.220 · 10−2
+1.96
+2.336 · 10−2
+1.57
+9.434 · 10−3
+1.41
+4.457 · 10−3
+1.14
+2.181 · 10−3
+1.03
+1.067 · 10−3
+1.06
+8.613 · 10−2
+8.448 · 10−2
+0.03
+4.565 · 10−2
+1.06
+1.335 · 10−2
+1.98
+3.689 · 10−3
+1.99
+9.205 · 10−4
+2.11
+2.280 · 10−4
+2.02
+5.777 · 10−5
+2.04
+2.720 · 10−2
+1.364 · 10−2
+1.13
+2.213 · 10−3
+3.13
+3.615 · 10−4
+2.91
+4.189 · 10−5
+3.34
+5.504 · 10−6
+3.08
+7.028 · 10−7
+2.97
+8.784 · 10−8
+3.1
+Table 1: Same as Figure 1 (left), but in tabular form.
+32
+
+r = 0
+r = 1
+r = 2
+h
+Error
+Order
+Error
+Order
+Error
+Order
+3.464 · 10−0
+1.850 · 10−0
+9.709 · 10−1
+4.999 · 10−1
+2.753 · 10−1
+1.358 · 10−1
+6.878 · 10−2
+7.869 · 10−2
+3.215 · 10−1
+-2.24
+1.132 · 10−1
+1.62
+4.152 · 10−2
+1.51
+1.838 · 10−2
+1.37
+8.733 · 10−3
+1.05
+4.296 · 10−3
+1.04
+1.359 · 10−1
+6.613 · 10−2
+1.15
+2.912 · 10−2
+1.27
+8.633 · 10−3
+1.83
+2.391 · 10−3
+2.15
+6.194 · 10−4
+1.91
+1.579 · 10−4
+2.01
+1.871 · 10−2
+4.133 · 10−2
+-1.26
+5.286 · 10−3
+3.19
+7.342 · 10−4
+2.97
+9.753 · 10−5
+3.38
+1.261 · 10−5
+2.89
+1.604 · 10−6
+3.03
+Table 2: Same as Figure 1 (right), but in tabular form.
+Israel formalism mentioned in Remark 3.9. EG was supported by NSF grant DMS-2012427. MN
+acknowledges support by the Austrian Science Fund (FWF) project F 65.
+A
+Optimal-order interpolation via averaging
+Below we verify that the interpolant described in Section 5 is an optimal-order interpolant in the
+sense of Remark 4.4, assuming that {Th}h>0 is shape-regular and quasi-uniform. Recall that quasi-
+uniformity means that maxT∈T N
+h h/hT is bounded above by a constant independent of h. In what
+follows, the letter C may depend on this constant as well as on the parameters N, hT /ρT , r, s, and
+t appearing below.
+Let ℓ(1), ℓ(2), . . . , ℓ(M) denote the canonical degrees of freedom for the Regge finite element space
+of degree r ≥ 0 on Th [21, Equation (2.4b)]. Each linear functional ℓ(i) is associated with a simplex
+D ∈ T k
+h of dimension k ≥ 1 in the following sense: ℓ(i) sends a symmetric (0, 2)-tensor field g to
+the integral of g|D against a (symmetric tensor-valued) polynomial of degree ≤ r − k + 1 over D.
+We enumerate these degrees of freedom with a local numbering system as follows.
+On a
+given N-simplex T ∈ T N
+h , the degrees of freedom associated with subsimplices of T are denoted
+ℓT
+1 , ℓT
+2 , . . . , ℓT
+MT . If T, T ′ ∈ T N
+h
+are two N-simplices with nonempty intersection, then it may happen
+that ℓT
+i and ℓT ′
+j
+coincide for some and i and j. We let S(i, T) denote the set of all pairs (j, T ′) for
+which ℓT
+i and ℓT ′
+j
+coincide.
+With the above local numbering system, let ψT
+1 , ψT
+2 , . . . , ψT
+MT denote the basis for the degree-r
+Regge finite element space that is dual to the above degrees of freedom. That is,
+ℓT
+i (ψT ′
+j ) =
+�
+1,
+if (j, T ′) ∈ S(i, T),
+0,
+otherwise.
+Let us assume that the degrees of freedom and basis functions above are first defined on a reference
+simplex and then transported to T via an affine transformation. A scaling argument shows that [21,
+Lemma 2.11]
+∥ψT
+i ∥Lp(T) ≤ ChN/p−2
+T
+(50)
+and
+|ℓT
+i (g)| ≤ Ch−N/p+2
+T
+∥g∥Lp(T)
+(51)
+for all g in the domain of ℓT
+i . Note that the −2 and the +2 appearing in the exponents above
+arise because of the way that pullbacks of (0, 2)-tensor fields behave under affine transformations;
+see [21, Lemma 2.11].
+33
+
+Let g be a symmetric (0, 2)-tensor field possessing W s,p(Ω)-regularity for every p ∈ [1, ∞] and
+every s > (N − 1)/p. The canonical interpolation operator Jh onto the Regge finite element space
+is defined elementwise by
+Jhg|T = J T
+h (g|T ) =
+MT
+�
+i=1
+ℓT
+i (g)ψT
+i .
+Let ¯gh denote the elementwise L2-projection of g onto the space of discontinuous piecewise
+polynomial symmetric (0, 2)-tensor fields of degree at most r. Since Jh is a projector, we have
+¯gh|T = J T
+h ( ¯gh|T ) =
+MT
+�
+i=1
+ℓT
+i (¯gh)ψT
+i .
+The interpolant discussed in Section 5 is defined by
+gh|T =
+MT
+�
+i=1
+
+
+1
+|S(i, T)|
+�
+(j,T ′)∈S(i,T)
+ℓT ′
+j (¯gh)
+
+ ψT
+i ,
+where |S(i, T)| denotes the cardinality of S(i, T).
+To analyze the error gh − g, let p ∈ [1, ∞], s ∈ ((N − 1)/p, r + 1], and t ∈ [0, s]. We have
+|gh − g|W t,p(T) ≤ |gh − Jhg|W t,p(T) + |Jhg − g|W t,p(T).
+The second term satisfies [21, Theorem 2.5]
+|Jhg − g|W t,p(T) ≤ Chs−t
+T
+|g|W s,p(T).
+(52)
+To bound the first term, we use the fact that
+ℓT
+i (g) =
+1
+|S(i, T)|
+�
+(j,T ′)∈S(i,T)
+ℓT ′
+j (g)
+to write
+(gh − Jhg)|T =
+MT
+�
+i=1
+1
+|S(i, T)|
+�
+(j,T ′)∈S(i,T)
+ℓT ′
+j (¯gh − g)ψT
+i .
+Using an inverse estimate, (50), (51), and a standard error estimate [14, Proposition 1.135] for the
+elementwise L2-projector, we obtain
+|gh − Jhg|W t,p(T) ≤ Ch−t
+T ∥gh − Jhg∥Lp(T)
+≤ Ch−t
+T
+�
+T ′:T ′∩T̸=∅
+h−N/p+2
+T ′
+∥¯gh − g∥Lp(T ′)hN/p−2
+T
+≤ Ch−t
+T
+�
+T ′:T ′∩T̸=∅
+∥¯gh − g∥Lp(T ′)
+≤ Ch−t
+T
+�
+T ′:T ′∩T̸=∅
+hs
+T ′|g|W s,p(T ′)
+≤ Chs−t
+T
+�
+T ′:T ′∩T̸=∅
+|g|W s,p(T ′).
+(53)
+Here, we have repeatedly used the fact that the ratio hT /hT ′ is bounded uniformly above and below
+by positive constants. Combining (52) and (53) shows that the error gh − g satisfies (28).
+34
+
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+

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@@ -0,0 +1,1502 @@
+Using a Penalized Likelihood to Detect Mortality
+Deceleration
+Silvio C. Patricio*1 and Trifon I. Missov1
+1The Interdisciplinary Centre on Population Dynamics, University of Southern Denmark
+Abstract
+We propose a novel method to detect deceleration in mortality patterns. For a gamma-
+Gompertz frailty model, we suggest maximizing a penalized likelihood in a Bayesian setting
+as an alternative to traditional likelihood inference and hypothesis testing. We compare the
+performance of the two methods on simulated and real mortality data.
+Keywords: Gompertz model; gamma-Gompertz model, mortality deceleration; penalized
+likelihood function; maximum a posteriori probability.
+1
+Introduction
+Human death-rate patterns are astoundingly log-linear over a wide range of adult ages. The
+Gompertz distribution (Gompertz, 1825) with an exponentially increasing hazard function cap-
+tures this accurately. The theory of unobserved heterogeneity and the associated frailty model
+(Vaupel et al., 1979) predicts a downward deviation at the oldest ages, to which only the most
+robust individuals in the population survive. Detecting such a deceleration in real data is not
+always successful (Gavrilova and Gavrilov, 2015; Newman, 2018), even though the vast major-
+ity of studies indicate that death rates at older ages increase at lower rates and can even level
+off (Curtsinger et al., 1992; Fukui et al., 1993, 1996; Carey et al., 1995; Khazaeli et al., 1998;
+Gampe, 2010, 2021; Rootz´en and Zholud, 2017; Alvarez et al., 2021; Camarda, 2022; Belzile
+et al., 2022). In a frailty model setting, testing for mortality deceleration is equivalent to testing
+whether the non-negative frailty parameter is strictly positive.
+Formally, denote by X a non-negative continuous random variable that describes individual
+human lifespans (complete or after a given adult age). If X ∼ Gompertz(a, b), where a is the
+mortality level at the initial age and b is the rate of aging, the associated hazard function (force
+of mortality) at time x
+µ(x) = lim
+ε↓0 P(x ≤ X < x + ε|X ≥ x).
+*silca@sam.sdu.dk
+1
+arXiv:2301.02853v1  [stat.ME]  7 Jan 2023
+
+is µ(x) = aebx . Vaupel et al. (1979) introduce a positive continuous random variable Z, called
+frailty, that acts multiplicatively on µ(x) and captures one’s unobserved susceptibility to death.
+The force of mortality for an individual with frailty Z = z is
+µ(x | Z = z) = z µ(x) .
+For a gamma-distributed frailty with E(Z) = 1 and VAR(Z) = σ2, the force of mortality of the
+population, i.e., the marginal hazard is
+¯µ(x) =
+aebx
+1 + σ2 a
+b (ebx − 1)
+(1)
+(see Vaupel et al. (1979) and Vaupel and Missov (2014) for all technicalities). Note that the
+variance of Z is often denoted by γ (e.g., in Vaupel and Missov, 2014) because it is also equal
+to the squared coefficient of variation of the distribution of frailty among survivors to any age x.
+If σ2 > 0, the force of mortality for the population ¯µ(x) starts deviating from the exponential
+pattern with increasing x and reaches an asymptote b/σ2. When σ2 = 0, i.e., when there is
+no unobserved heterogeneity, the model for the population reduces to the (Gompertz) model for
+individuals with an exponentially increasing hazard function µ(x) = aebx.
+Testing for mortality deceleration in this setting reduces to statistical testing whether σ2 = 0
+given the alternative σ2 > 0. The frailty parameter σ2 can take a value on the boundary of
+the parameter space (σ2 = 0). This violates the standard underlying assumptions about the
+asymptotic properties of likelihood-based inference and statistical hypothesis testing (see, for
+example, B¨ohnstedt and Gampe, 2019). As a result, the asymptotic distribution of the maximum
+likelihood estimator may not be Gaussian.
+In this paper, we treat the problem of identifying whether σ2 > 0 or σ2 = 0 as a model
+misspecification problem, i.e., we consider the gamma-Gompertz model when it is the Gompertz
+model that actually holds. In this setting, we suggest subtracting a penalty from the log-likelihood
+function. This penalty will be responsible for shrinking σ2 to zero when there is no heterogeneity,
+as well as for adding a small bias to the Maximum Likelihood Estimator (MLE) when the effect
+of unobserved heterogeneity is non-negligible. We carry out Monte Carlo simulation experiments
+to evaluate the accuracy and precision of the estimates obtained by maximizing the likelihood
+function, on the one hand, and the penalized likelihood function, on the other.
+In Section 2, we formulate the model misspecification problem and introduce inference
+methodology taking advantage of the maximum a posteriori probability (MAP). Then we carry
+out a Monte Carlo simulation study to compare the performance of maximizing a standard and a
+penalized likelihood. In Section 3, we compare the latter on mortality data for France, Japan and
+the USA. Section 4 discusses the advantages and drawbacks of applying our method to detect
+heterogeneity (deceleration) in mortality patterns.
+2
+Methodology
+Suppose X is a random sample with a cumulative distribution function G(x), and we fit the
+incorrect family of densities {f(x; θ), θ ∈ Θ} to the data using MLE. The misspecified log-
+likelihood is
+2
+
+ℓ(θ; X) =
+n
+�
+i=1
+log f(Xi; θ).
+Applying the law of large numbers, we get in the limit what the misspecified log-likelihood
+function ℓ(θ; X) looks like for each θ ∈ Θ (see the right-hand side below):
+1
+nℓ(θ; X) = 1
+n
+n
+�
+i=1
+log f(Xi; θ)
+a.s
+−→ Eg (log f(X1; θ)) =
+�
+Im(X1)
+log f(x; θ)dG(x) .
+(2)
+Assume there is no heterogeneity in the data (σ2 = 0), and we fit a gamma-Gompertz model.
+In other words, we observe an exponential death-rate increase in the data, but we estimate a
+model that implies a downward deviation from the exponential at the oldest ages. As shown in
+(2), we will estimate σ2 close to but never equal to zero.
+In this model setting, the standard technique is to estimate both the Gompertz and the gamma-
+Gompertz models and compare their goodness of fit. However, minor changes in the data can
+result in different models being selected, which can reduce prediction accuracy and lead to mis-
+interpretations about the mortality deceleration and the mortality plateau. B¨ohnstedt and Gampe
+(2019) derive the asymptotic distribution of the likelihood ratio test statistic to detect heterogene-
+ity. Here, we would like to suggest an alternative that does not involve hypothesis testing. Using
+the latter has been widely discussed and rethought in the Statistics community (Berk et al., 2010;
+Head et al., 2015; Vidgen and Yasseri, 2016; Bruns and Ioannidis, 2016), especially in relation
+to the arbitrary choice of the α-level (most often 0.1, 0.05, or 0.01) and sample size issues.
+Maximum likelihood estimators, obtained by maximizing the log-likelihood function, often
+have low bias and large variance. Estimation accuracy can sometimes be improved by shrinking
+some parameters to zero (Tibshirani, 1996). The associated shrinkage estimator improves the
+overall prediction accuracy at the expense of introducing a small bias to reduce the variance of
+the parameters. This class of estimators is implicit in Bayesian inference and penalized likelihood
+inference. Using shrinkage estimators is applied as an alternative to hypothesis testing. Lasso,
+Ridge and Stein-type estimators are the most widely used examples of penalizing methods (see,
+for example, Hastie et al., 2009).
+2.1
+Inference
+Let Dx be the number of deaths in a given age interval [x, x + 1) for x = 0, . . . , m, and Ex
+denote the number of person-years lived in the same interval (see, for example Brillinger, 1986;
+Macdonald et al., 2018). Define D = (D0, D1, . . . , Dm)⊤ and E = (E0, E1, . . . , Em)⊤. In
+addition, let θ = (a, b, σ2)⊤ ∈ Θ be the parameter vector that characterizes the force of mortality
+at age x of the gamma-Gompertz model given by (1).
+Assume Dx are Poisson-distributed with E(Dx) = VAR(Dx) = µ(x; θ)Ex for x = 0, . . . , m
+(Brillinger, 1986). Under this assumption, the log-likelihood function for θ = (a, b, σ2)⊤ is
+given by
+ℓ(θ) = ℓ(θ|D, E) =
+m
+�
+x=0
+[Dx ln µ(x; θ)) − Ex µ(x; θ)] .
+(3)
+3
+
+Maximizing ℓ(θ) with respect to θ = (a, b, σ2)⊤ yields the maximum-likelihood (ML) estimate
+ˆθ.
+Let us now define a penalized log-likelihood function as
+ℓp(θ) = ℓ(θ) − p(σ2) ,
+(4)
+where ℓ(θ) is the standard log-likelihood (3), while p(σ2) is a penalty function. The penalized
+maximum-likelihood estimate is obtained by maximizing ℓp(θ) with respect to θ = (a, b, σ2)⊤.
+For the problem addressed in this paper, the penalty function p(σ2) must be a non-decreasing
+monotonic continuous function and lim
+σ2↓0 p(σ2) > p(σ2) for all σ2 > 0.
+In a Bayesian framework, maximizing (4) is equivalent to maximizing a posterior distribution
+in a setting, in which e−p(σ2)/Cp, Cp :=
+�
+Θ e−p(σ2)∇θ < ∞, is taken as a prior distribution of
+θ. This procedure yields the maximum a posteriori probability (MAP) estimator. MAP is the
+only Bayesian estimator that minimizes the expected canonical loss (Pereyra, 2019) and is widely
+used in image and video processing (Greig et al., 1989; Afonso et al., 2010; Belekos et al., 2010).
+As σ2 describes the variance of frailty at the starting age of analysis, the standard approach
+would be to specify an inverse gamma prior distribution for it (Gelman et al., 1995). The inverse
+gamma distribution is heavy-tailed and keeps probability mass further from zero than the gamma
+distribution. In addition, while the inverse-gamma mode is always positive, the gamma mode
+can also be zero (Llera and Beckmann, 2016). As we aim to test whether σ2 = 0 or σ2 > 0, we
+will use the log-kernel of the gamma distribution to define the penalty function as
+p(σ2) = λ
+�
+σ2 + ln σ2�
+(5)
+for some non-negative λ. When λ < 1, using (5) is equivalent to specifying a gamma prior
+distribution for σ2 with parameters α = 1 − λ and β = λ.
+When m → ∞, the effect of the penalty diminishes regardless of the size of λ. For human
+life table data m is finite, thus λ ≥ 0 is a constant that controls the relative impact of the penalty
+function on the estimates. When λ = 0, the penalty term has no effect, and maximizing the
+penalized likelihood will produce the standard maximum likelihood estimates (MLE). However,
+as λ → ∞, the impact of the penalty grows, and the maximum penalized likelihood estimates
+for σ2 will approach zero, providing high precision, but low accuracy.
+Choosing λ is sensible in a wide range of applications (Li et al., 2009; Bhattacharya and
+McNicholas, 2014). Therefore, in accordance with the recommendations in Li et al. (2009),
+we carry out a pilot simulation study, in which we find that choosing λ = 1
+2 provides similar
+precision to the one by MLE when σ2 > 0, but better accuracy and precision when σ2 = 0
+(simulation results are presented in the next subsection). As a result, the final expression for the
+penalized log-likelihood we propose is
+ℓp(θ) =
+m
+�
+x=0
+[Dx ln µ(x; θ) − Ex µ(x; θ)] − 1
+2
+�
+ln σ2 + σ2�
+.
+(6)
+From a Bayesian perspective, choosing λ = 1
+2 provides an informative prior distribution for
+σ2. As for human populations we are likely to estimate σ2 < 1 (Missov, 2013), the specified
+prior will provide for σ2 a distribution with a mode equal to zero, a median equal to 0.4549, and
+4
+
+0.00000
+0.00010
+0.00020
+0.00030
+−600
+−400
+−200
+0
+Parameter a
+a
+log−likelihood
+MLE
+MAP
+a = 1e−04
+b = 0.1
+σ2 = 0.1
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+−8000
+−4000
+0
+Parameter b
+b
+log−likelihood
+MLE
+MAP
+a = 1e−04
+b = 0.1
+σ2 = 0.1
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+−5.5
+−4.5
+−3.5
+Parameter σ2
+σ2
+log−likelihood
+MLE
+MAP
+a = 1e−04
+b = 0.1
+σ2 = 0.1
+0.00000
+0.00010
+0.00020
+0.00030
+−800
+−600
+−400
+−200
+0
+Parameter a
+a
+log−likelihood
+MLE
+MAP
+a = 1e−04
+b = 0.1
+σ2 = 0
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+−8000
+−4000
+0
+Parameter b
+b
+log−likelihood
+MLE
+MAP
+a = 1e−04
+b = 0.1
+σ2 = 0
+0.000
+0.005
+0.010
+0.015
+0.020
+0.025
+0.030
+−3.55
+−3.45
+−3.35
+Parameter σ2
+σ2
+log−likelihood
+MLE
+MAP
+a = 1e−04
+b = 0.1
+σ2 = 0
+Figure 1: Plots of the profile log-likelihood and penalized log-likelihood functions of the param-
+eters. In the first row we used synthetic data from a gamma-Gompertz model with parameters
+a = 0.0001, b = 0.1 and σ2 = 0.1, in the second row we from a Gompertz model with parameters
+a = 0.0001 and b = 0.1.
+mean equal to 1. Furthermore, the prior provides a probability mass of 0.6826 in the interval
+(0, 1].
+Figure 1 shows the log-likelihood and penalized log-likelihood functions for all parameters
+when σ2 > 0 (first row) and σ2 = 0 (second row). When σ2 > 0, the penalty function affects
+neither the shape of the log-likelihood, nor the location of its maximum. However, when σ2 = 0,
+adding a penalty yields a higher maximum at 0. Moreover, when σ2 = 0, the first and second
+derivatives of the penalized log-likelihood are higher than their respective counterparts of the
+log-likelihood. As a result, derivative-based optimization methods may reach the maximum
+point faster, and the estimator ˆσ2 may have a smaller variance.
+2.2
+Monte Carlo simulations
+We carry out Monte Carlo simulations to explore the performance of the MAP and ML methods
+in estimating the gamma-Gompertz model parameters. We use the R software (Team et al.,
+2022) to maximize the log-likelihood and the penalized log-likelihood functions via the optim
+function applying as a pre-step differential evolution (Storn and Price, 1997; Ardia et al., 2011).
+The performance of the ML and MAP estimators are evaluated by calculating two measures: the
+bias and the standard deviation.
+We generate 10,000 random samples from this model for some parameter values (scenarios
+with sample sizes of 2,000 and 5,000 were also considered, and are presented in the appendix).
+From these samples, we generate life tables and use them to estimate model parameters via the
+5
+
+MAP and MLE methods. This process was repeated 2,000 times. In the presence of unobserved
+heterogeneity, the true parameter values are a1 = 0.0001 and a2 = 0.00001 for a, b1 = 0.1 and
+b2 = 0.15 for b, and σ2
+1 = 0.2 and σ2
+2 = 0.8 for σ2. When there is no heterogeneity (σ2 = 0),
+the true parameter values are a1 = 0.0001, a2 = 0.0003 and a3 = 0.0005 for a, and b1 = 0.09,
+b2 = 0.10 and b3 = 0.11 for b.
+Table 1: Simulation results: gamma-Gompertz model and sample size 10,000.
+There is heterogeneity
+MLE estimator
+MAP estimator
+Bias
+Standard deviation
+Bias
+Standard deviation
+Parameter
+a
+b
+σ2
+a
+b
+σ2
+a
+b
+σ2
+a
+b
+σ2
+(a1, b1, σ2
+1)
+0.000053
+-0.000051
+-0.000223
+0.000051
+0.001499
+0.020721
+0.000055
+-0.000134
+-0.001626
+0.000052
+0.001502
+0.020791
+(a1, b1, σ2
+2)
+0.000060
+-0.000292
+-0.007787
+0.000056
+0.001784
+0.035822
+0.000061
+-0.000354
+-0.009229
+0.000056
+0.001783
+0.035795
+(a1, b2, σ2
+1)
+0.000077
+0.000131
+0.004431
+0.000056
+0.002181
+0.020569
+0.000080
+0.000015
+0.003096
+0.000057
+0.002186
+0.020631
+(a1, b2, σ2
+2)
+0.000085
+-0.000262
+-0.003547
+0.000061
+0.002557
+0.034714
+0.000087
+-0.000348
+-0.004920
+0.000061
+0.002556
+0.034687
+(a2, b1, σ2
+1)
+0.000007
+-0.000349
+-0.003349
+0.000008
+0.001315
+0.019464
+0.000008
+-0.000417
+-0.004597
+0.000008
+0.001318
+0.019523
+(a2, b1, σ2
+2)
+0.000009
+-0.000635
+-0.013592
+0.000008
+0.001515
+0.032551
+0.000009
+-0.000683
+-0.014810
+0.000008
+0.001514
+0.032528
+(a2, b2, σ2
+1)
+0.000009
+-0.000170
+0.002295
+0.000008
+0.001963
+0.019377
+0.000009
+-0.000268
+0.001078
+0.000008
+0.001966
+0.019430
+(a2, b2, σ2
+2)
+0.000011
+-0.000650
+-0.007809
+0.000009
+0.002273
+0.032534
+0.000011
+-0.000721
+-0.009014
+0.000009
+0.002272
+0.032510
+There is no heterogeneity
+MLE estimator
+MAP estimator
+Bias
+Standard deviation
+Bias
+Standard deviation
+Parameter
+a
+b
+σ2
+a
+b
+σ2
+a
+b
+σ2(10−16)
+a
+b
+σ2(10−15)
+(a1, b1, σ2)
+0.000005
+-0.000055
+0.000181
+0.000006
+0.000875
+0.008460
+0.000007
+-0.000239
+0.125942
+0.000006
+0.000791
+0.6308937
+(a1, b2, σ2)
+0.000006
+-0.000050
+0.000222
+0.000006
+0.000968
+0.008907
+0.000007
+-0.000407
+0.060433
+0.000006
+0.000870
+0.1460452
+(a1, b3, σ2)
+0.000006
+-0.000057
+0.000428
+0.000006
+0.001082
+0.009726
+0.000008
+-0.000305
+0.089035
+0.000006
+0.000978
+8.212895
+(a2, b1, σ2)
+0.000011
+0.000138
+0.001864
+0.000016
+0.000913
+0.009010
+0.000016
+-0.000198
+0.004470
+0.000015
+0.000811
+0.294978
+(a2, b2, σ2)
+0.000014
+0.000117
+0.001094
+0.000016
+0.001042
+0.009873
+0.000019
+-0.000264
+0.003661
+0.000015
+0.000859
+4.922260
+(a2, b3, σ2)
+0.000015
+0.000164
+0.002204
+0.000016
+0.001150
+0.010369
+0.000022
+-0.000216
+0.007167
+0.000016
+0.001017
+3.551245
+(a3, b1, σ2)
+0.000018
+0.000143
+0.001062
+0.000025
+0.000979
+0.009712
+0.000027
+-0.000138
+0.001124
+0.000025
+0.000876
+1.898573
+(a3, b2, σ2)
+0.000024
+0.000076
+0.000505
+0.000025
+0.001057
+0.009721
+0.000030
+-0.000112
+0.001177
+0.000025
+0.000942
+8.114498
+(a3, b3, σ2)
+0.000025
+0.000117
+0.001427
+0.000025
+0.001178
+0.009999
+0.000033
+-0.000171
+0.001415
+0.000024
+0.000990
+0.000025
+The simulation results are presented in Table 1. In the presence of unobserved heterogeneity,
+both methods underestimate b and σ2. They also introduce a small positive bias to a, the one pro-
+vided by ML estimator being slightly smaller. However, in general the ML and MAP estimators
+perform equally well, with a similar bias and standard deviation.
+In the absence of unobserved heterogeneity, the ML estimator provides again a smaller bias
+for a and b than the MAP estimator. However, in this case, the MAP method estimates more
+precisely the frailty parameter σ2, with a bias and a standard deviation close to zero (∝ 10−15).
+The MAP estimator also provides a slight reduction in the standard deviation of parameter b.
+By the Monte Carlo simulation we also calculate the proportion of trials in which MAP
+estimates σ2 > 0 when the true values is σ2 = 0 (error type I), as well as the proportion of trials
+in which MAP estimates σ2 = 0 when the true values is σ2 > 0 (error type II). Based on our
+simulations, the type I errro equals 0.001502, while the type II error is 0.001126.
+The Monte Carlo simulations show that using a penalizing likelihood function (6) is an alter-
+native to hypothesis testing, the latter being dependent on the asymptotic distribution of the ML
+estimator, sample size and the arbitrary choice of the α-level (B¨ohnstedt and Gampe, 2019).
+3
+Performance of MAP and ML estimators on HMD data
+In this section, we estimate the gamma-Gompertz model via ML and MAP using mortality data
+from the Human Mortality Database (HMD, 2022). We take exposures and raw death counts for
+6
+
+the female population of France, Japan and the USA in the years 1960, 1980, 2000, and 2020,
+after age 70. We apply again R (Team et al., 2022) to compute the ML and MAP estimates of
+θ = (a, b, σ2)′ by using differential evolution. We use the mean squared error given by
+MSE = 1
+n
+m
+�
+x=0
+�
+ln mx − ln ¯µ(x; ˆθ)
+�2
+,
+to assess the goodness of fit.
+Table 2: Life expectancy: gamma Gompertz–Makeham model and ML estimates.
+ML Estimates
+MAP Estimates
+Country
+Year
+a
+b
+σ2
+MSE
+a
+b
+σ2
+MSE
+France
+1960
+0.003582
+0.107599
+0.016726
+0.100588
+0.003593
+0.107483
+0.015902
+0.100540
+1980
+0.002210
+0.112032
+0.003393
+0.045776
+0.002220
+0.111729
+0.003117
+0.046747
+2000
+0.001247
+0.117749
+0.000001
+0.073648
+0.001250
+0.117689
+0
+0.073262
+2020
+0.000957
+0.119494
+0.000002
+0.094243
+0.000960
+0.119416
+0
+0.093697
+Japan
+1960
+0.004782
+0.105858
+0.039989
+0.011366
+0.004782
+0.105845
+0.039593
+0.011493
+1980
+0.002009
+0.117886
+0.015251
+0.053944
+0.002009
+0.117941
+0.015942
+0.053328
+2000
+0.001140
+0.115268
+0.000015
+0.064604
+0.001142
+0.115118
+0
+0.063728
+2020
+0.000575
+0.125814
+0.000233
+0.104944
+0.000574
+0.125870
+0.000225
+0.105597
+USA
+1960
+0.004701
+0.095797
+0.032146
+0.111711
+0.004699
+0.095814
+0.032802
+0.110740
+1980
+0.003612
+0.093688
+0.000001
+0.054483
+0.003609
+0.093720
+0
+0.054652
+2000
+0.002712
+0.100566
+0.000003
+0.030967
+0.002714
+0.100540
+0
+0.030873
+2020
+0.002473
+0.101652
+0.000001
+0.022735
+0.002476
+0.101612
+0
+0.022618
+Table 2 shows the results of applying ML and MAP methods to the datasets described above.
+The MAP estimator provides lower MSEs in 8 of the 12 datasets. When the standard ML method
+estimates σ2 < 10−4, our novel method estimates σ2 = 0 and provides a smaller MSE. This
+suggests that the MAP provides a slightly better fit to the data. Overall, MAP performs better
+than ML when unobserved heterogeneity is not detected, and while for estimates of ˆσ2 > 0 ML
+has a slight advantage.
+The results from the real-data application back up the results from the Monte Carlo simula-
+tions in Section 2. In the presence of unobserved heterogeneity, the MLE method provides the
+most precise and accurate estimates. The MAP method, though, has just slightly lower precision.
+On the other hand, in the absence of unobserved heterogeneity, the MAP provides smaller bias
+and variance in its estimates compared to MLE.
+3.1
+Examples when MAP and ML estimators yield different outcomes
+Using MAP and ML estimators does not always lead to the same statistical inference. One of
+them can detect heterogeneity in cases when the other does not. We will illustrate this on HMD
+data for the Japanese female population in 2009 and the French female population born in 1848,
+ages 70+. To assess the goodness of fit, we will use again MSE.
+For Japanese females in 2009, ML yields estimates ˆθMLE = (0.006359, 0.133805, 0.070513)′
+with standard errors SE(a) = 0.000188, SE(b) = 0.002263 and SE(σ2) = 0.021156. The 95%
+confidence interval for σ2 is (0.029047, 0.111978) indicating stasitically significant unobserved
+7
+
+heterogeneity, i.e., the existence of mortality deceleration. On the other hand, the MAP method
+estimates ˆθMAP = (0.006966, 0.125440, 0)′, indicating the absence of unobserved heterogeneity.
+Comparing the goodness of fit of both methods speaks in favor of the MAP outcome: MAP’s
+MSE is by 37% lower than ML’s LSE (0.018691 for MAP vs 0.029958 for ML). It indicates that
+unobserved heterogeneity is negligible and that the gamma-Gompertz model is misspecified.
+70
+80
+90
+100
+110
+−5
+−4
+−3
+−2
+−1
+0
+Age
+log−force of mortality
+Mortality rate
+MLE
+MAP
+Japan
+70
+75
+80
+85
+90
+95
+100
+−3.0
+−2.0
+−1.0
+0.0
+Age
+log−force of mortality
+Mortality rate
+MLE
+MAP
+France
+Figure 2: MAP and MLE estimates of the force of mortality for the Japanese population in 2009
+and the Swedish population born in 1881, after age 70
+The left panel of Figure 2 shows that both methods estimate a similar logarithmic force of
+mortality at most ages. However, after age 100, the MLE deviates downward from the observed
+logarithmic death rates.
+The MAP also provides a better fit and different conclusion for the cohort of French females
+born in 1848. While ML estimates ˆθMLE = (0.053748, 0.090552, 0.008604)′ with SE(a) =
+0.000317, SE(b) = 0.001273, SE(σ2) = 0.007562 and provides an MSE equal 0.046222,
+MAP estimates ˆθMAP = (0.053113, 0.094921, 0.036466)′ and provides MSE = 0.034226, i.e.,
+MAP’s MSE is by 26% smaller than ML’s MSE.
+Furthermore, while the MAP estimate of σ2 suggests that there is non-negligible unobserved
+heterogeneity, the ML estimate and standard error for σ2 indicates the opposite: the amount
+of unobserved heterogeneity is not statistically significant. The right panel of Figure 2 shows
+the difference between these estimates. MAP’s estimate shows a leveling-off in the force of
+mortality, while the MLE shows a log-linear increase in the hazard function.
+4
+Concluding remarks
+B¨ohnstedt and Gampe (2019) introduced a formal procedure to identify whether σ2 > 0 or
+σ2 = 0 in a hypothesis testing setting: they studied the asymptotic properties of the maximum
+likelihood estimator and the likelihood ratio test (LRT) for H0 : σ2 = 0 vs. H1 : σ2 = 0 for
+8
+
+the gamma-Gompertz model. However, LRTs are based on the asymptotic distribution of the
+maximum likelihood estimator, hence its convergence depends on the sample size. Moreover,
+conclusions drawn from hypothesis tests are dependent on the arbitrary choice of the significance
+level or p-value.
+We suggest an alternative method by considering the problem as model misspecification. We
+add a penalty function to the likelihood so that we make sure that ˆσ2 = 0 when there is no het-
+erogeneity. We also present a Bayesian interpretation (MAP) to our method. We take advantage
+of robust Monte Carlo simulations to measure the bias and standard deviation of the ML and
+MAP methods in scenarios with and without unobserved heterogeneity. We also compare the
+performance of both methods for estimating the gamma-Gompertz model parameters using ac-
+tual mortality data from the Human Mortality Database. The two methods work almost equally
+well, the ML having a slight advantage, in the presence of unobserved heterogeneity. However,
+in the absence of the latter, the MAP method provides an estimate closer to 0 (ˆσ2 ≈ 10−20) and a
+better fit to the model in comparison to ML. As a result, the method we propose here can be used
+as an alternative to likelihood ratio testing for the gamma-Gompertz model with H0 : σ2 = 0 vs.
+H1 : σ2 > 0. On the one hand, the MAP method does not depend on any asymptomatic distribu-
+tion, its performance is not strongly affected by sample size, and it also does not depend on the
+arbitrary choice of the significance level. On the other hand, MAP provides similar estimates to
+the ones by ML when σ2 > 0 and more accurate estimates when σ2 = 0.
+Acknowledgments
+The research leading to this publication is a part of a project that has received funding from
+the European Research Council (ERC) under the European Union’s Horizon 2020 research and
+innovation programme (Grant agreement No. 884328 – Unequal Lifespans). Silvio C. Patricio
+gratefully acknowledges the support provided from AXA Research Fund, through the funding
+for the “AXA Chair in Longevity Research”.
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+Appendices
+Table 3: Simulation results: gamma-Gompertz model and sample size 2,000.
+There is heterogeneity
+MLE estimator
+MAP estimator
+Bias
+Standard deviation
+Bias
+Standard deviation
+Parameter
+a
+b
+σ2
+a
+b
+σ2
+a
+b
+σ2
+a
+b
+σ2
+(a1, b1, σ2
+1)
+0.000089
+-0.001172
+-0.020361
+0.000111
+0.003286
+0.047684
+0.000104
+-0.001709
+-0.029261
+0.000117
+0.003465
+0.051082
+(a1, b1, σ2
+2)
+0.000110
+-0.002075
+-0.048634
+0.000128
+0.003972
+0.078621
+0.000118
+-0.002397
+-0.055984
+0.000129
+0.003970
+0.078467
+(a1, b2, σ2
+1)
+0.000104
+-0.000906
+-0.008883
+0.000124
+0.004826
+0.047186
+0.000120
+-0.001603
+-0.016820
+0.000130
+0.005032
+0.049798
+(a1, b2, σ2
+2)
+0.000124
+-0.001978
+-0.029154
+0.000139
+0.005738
+0.077069
+0.000133
+-0.002420
+-0.036104
+0.000140
+0.005733
+0.076901
+(a2, b1, σ2
+1)
+0.000030
+-0.003810
+-0.046694
+0.000021
+0.003024
+0.045092
+0.000034
+-0.004407
+-0.057147
+0.000024
+0.003388
+0.052275
+(a2, b1, σ2
+2)
+0.000038
+-0.005056
+-0.097144
+0.000023
+0.003338
+0.070146
+0.000039
+-0.005316
+-0.103429
+0.000024
+0.003333
+0.069973
+(a2, b2, σ2
+1)
+0.000027
+-0.003998
+-0.030152
+0.000021
+0.004259
+0.043531
+0.000030
+-0.004673
+-0.038167
+0.000022
+0.004481
+0.046813
+(a2, b2, σ2
+2)
+0.000032
+-0.005191
+-0.065676
+0.000025
+0.005158
+0.075375
+0.000034
+-0.005565
+-0.071836
+0.000025
+0.005146
+0.075141
+There is no heterogeneity
+MLE estimator
+MAP estimator
+Bias
+Standard deviation
+Bias
+Standard deviation
+Parameter
+a
+b
+σ2
+a
+b
+σ2
+a
+b
+σ2(10−12)
+a
+b
+σ2(10−12)
+(a1, b1, σ2)
+0.000015
+-0.001225
+0.000002
+0.000013
+0.001615
+0.006546
+0.000016
+-0.001341
+0.021800
+0.000014
+0.001693
+0.000287
+(a1, b2, σ2)
+0.000014
+-0.001224
+0.000002
+0.000013
+0.001833
+0.007389
+0.000016
+-0.001312
+0.018541
+0.000014
+0.001858
+0.004010
+(a1, b3, σ2)
+0.000014
+-0.001323
+0.000002
+0.000013
+0.001998
+0.008857
+0.000015
+-0.001324
+0.024195
+0.000015
+0.002098
+0.001130
+(a2, b1, σ2)
+0.000028
+-0.000591
+0.000003
+0.000031
+0.001703
+0.009242
+0.000031
+-0.000983
+0.000923
+0.000033
+0.001666
+0.000003
+(a2, b2, σ2)
+0.000029
+-0.000837
+0.000003
+0.000031
+0.001857
+0.010440
+0.000031
+-0.000842
+0.000731
+0.000031
+0.001804
+0.000009
+(a2, b3, σ2)
+0.000030
+-0.000810
+0.000004
+0.000031
+0.002067
+0.011604
+0.000035
+-0.001196
+0.000701
+0.000034
+0.002086
+0.000008
+(a3, b1, σ2)
+0.000034
+-0.000398
+0.000003
+0.000046
+0.001739
+0.011618
+0.000037
+-0.000606
+0.000220
+0.000048
+0.001681
+0.000001
+(a3, b2, σ2)
+0.000036
+-0.000503
+0.000005
+0.000049
+0.002062
+0.013431
+0.000040
+-0.000670
+0.000151
+0.000048
+0.001901
+0.000002
+(a3, b3, σ2)
+0.000040
+-0.000238
+0.000008
+0.000050
+0.002247
+0.014168
+0.000045
+-0.000722
+0.000108
+0.000051
+0.002061
+0.000008
+Table 4: Simulation results: gamma-Gompertz model and sample size 5,000.
+There is heterogeneity
+MLE estimator
+MAP estimator
+Bias
+Standard deviation
+Bias
+Standard deviation
+Parameter
+a
+b
+σ2
+a
+b
+σ2
+a
+b
+σ2
+a
+b
+σ2
+(a1, b1, σ2
+1)
+0.000067
+-0.000379
+-0.004470
+0.000071
+0.002078
+0.029046
+0.000071
+-0.000552
+-0.007370
+0.000072
+0.002090
+0.029281
+(a1, b1, σ2
+2)
+0.000070
+-0.000582
+-0.015178
+0.000077
+0.002506
+0.049888
+0.000074
+-0.000708
+-0.018078
+0.000077
+0.002504
+0.049808
+(a1, b2, σ2
+1)
+0.000090
+-0.000273
+0.001254
+0.000077
+0.002969
+0.028061
+0.000095
+-0.000509
+-0.001483
+0.000078
+0.002982
+0.028253
+(a1, b2, σ2
+2)
+0.000094
+-0.000514
+-0.007569
+0.000083
+0.003583
+0.048230
+0.000097
+-0.000687
+-0.010324
+0.000084
+0.003579
+0.048152
+(a2, b1, σ2
+1)
+0.000014
+-0.001320
+-0.014568
+0.000011
+0.001786
+0.026808
+0.000014
+-0.001465
+-0.017208
+0.000011
+0.001795
+0.027016
+(a2, b1, σ2
+2)
+0.000015
+-0.001650
+-0.033574
+0.000013
+0.002344
+0.049087
+0.000016
+-0.001748
+-0.036030
+0.000013
+0.002342
+0.049016
+(a2, b2, σ2
+1)
+0.000014
+-0.001299
+-0.006177
+0.000012
+0.002706
+0.026681
+0.000015
+-0.001505
+-0.008713
+0.000012
+0.002717
+0.026863
+(a2, b2, σ2
+2)
+0.000015
+-0.001571
+-0.019479
+0.000014
+0.003352
+0.046735
+0.000016
+-0.001714
+-0.021904
+0.000014
+0.003349
+0.046666
+There is no heterogeneity
+MLE estimator
+MAP estimator
+Bias
+Standard deviation
+Bias
+Standard deviation
+Parameter
+a
+b
+σ2
+a
+b
+σ2
+a
+b
+σ2(10−12)
+a
+b
+σ2(10−12)
+(a1, b1, σ2)
+0.000008
+-0.000312
+0.000009
+0.000008
+0.001139
+0.006564
+0.000008
+-0.000441
+0.024410
+0.000009
+0.001142
+0.000058
+(a1, b2, σ2)
+0.000008
+-0.000339
+0.000011
+0.000009
+0.001290
+0.007315
+0.000009
+-0.000497
+0.024494
+0.000009
+0.001259
+0.000179
+(a1, b3, σ2)
+0.000008
+-0.000237
+0.000011
+0.000009
+0.001398
+0.007896
+0.000009
+-0.000483
+0.037635
+0.000009
+0.001417
+0.000985
+(a2, b1, σ2)
+0.000015
+0.000055
+0.000823
+0.000021
+0.001212
+0.008106
+0.000016
+-0.000102
+0.001137
+0.000022
+0.001184
+0.000012
+(a2, b2, σ2)
+0.000017
+0.000028
+0.000732
+0.000022
+0.001363
+0.009211
+0.000020
+-0.000228
+0.001954
+0.000023
+0.001319
+0.001225
+(a2, b3, σ2)
+0.000019
+0.000122
+0.000182
+0.000022
+0.001473
+0.009371
+0.000021
+-0.000024
+0.002634
+0.000022
+0.001387
+0.000014
+(a3, b1, σ2)
+0.000023
+0.000212
+0.001357
+0.000033
+0.001265
+0.009088
+0.000023
+-0.000068
+0.000266
+0.000033
+0.001200
+0.000001
+(a3, b2, σ2)
+0.000025
+0.000198
+0.000845
+0.000034
+0.001391
+0.009982
+0.000030
+-0.000154
+0.000558
+0.000035
+0.001332
+0.000048
+(a3, b3, σ2)
+0.000026
+0.000127
+0.000722
+0.000035
+0.001590
+0.011029
+0.000034
+-0.000151
+0.001303
+0.000035
+0.001464
+0.003727
+12
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+Reconfigurable microresonators induced in side-coupled optical fibers 
+V. VASSILIEV AND M. SUMETSKY*  
+Aston Institute of Photonic Technologies, Aston University, Birmingham B4 7ET, UK 
+*Email:  m.sumetsky@aston.ac.uk 
+ 
+ 
+We experimentally demonstrate that side-coupling of coplanar bent optical fibers can induce a high Q-factor whispering 
+gallery mode (WGM) optical microresonator. To explain the effect, we consider WGMs with wavelengths close to the cutoff 
+wavelengths (CWs) of these fibers which slowly propagate along the fiber axes. In the vicinity of the touching region, WGMs 
+of adjacent fibers are coupled to each other, and CWs experience sub-nanoscale axial variation proportional to the coupling 
+strength. We show that in certain cases the CW variation leads to full localization of the WGMs and the creation of an optical 
+microresonator. By varying the characteristic curvature fiber radius from the centimeter order to millimeter order, we 
+demonstrate fully mechanically reconfigurable high Q-factor optical microresonators with dimensions varying from the 
+millimeter order to 100-micron order and free spectral range varying from a picometer to hundreds of picometers. The new 
+microresonators may find applications in cavity QED, microresonator optomechanics, frequency comb generation with 
+tunable repetition rate, tunable lasing, and tunable processing and delay of optical pulses.  
+ 
+ 
+1. Introduction 
+Microphotonic devices and circuits commonly consist of one or 
+multiple connected basic elements, such as waveguides, couplers, 
+and ring resonators [1, 2]. In addition to the requirements of high 
+fabrication precision and low losses [2, 3], the tunability of these 
+circuits and devices is of critical importance for a variety of 
+applications [4, 5]. While more complex tunable microphotonics 
+circuits are targeted at tunability enabling quite arbitrary 
+predetermined signal processing (see e.g., [1]), simple microdevices, 
+such as standing along tunable three-dimensional microresonators, 
+allow for unique functionalities not possible to achieve by other 
+means. For a variety of applications, the tunability of spherical, 
+toroidal, and bottle microresonators has been demonstrated using 
+mechanical stretching, heating, and nonlinear light effects including 
+those in monolithic and specially coated microresonators [6-10]. In 
+most of these approaches, it is only possible to tune series of 
+wavelength eigenvalues simultaneously without noticeable change 
+in their separation.  
+ However, for several applications, which include cavity QED [8, 
+11, 12], optomechanics [13, 14], frequency microcomb generation 
+[15, 16], optical signal processing and delay [4, 5, 17], and lasing 
+[18-21], it is critical to have microresonators with tunable 
+eigenwavelength separation. For example, the latter allows the 
+creation of optical frequency microcomb generators and microlasers 
+with continuously tunable repetition rate and wavelength and to tune 
+the microresonator eigenfrequency separation in resonance with the 
+frequency of its mechanical oscillations. Considerable variation of 
+the eigenwavelength separation commonly requires the variation of 
+microresonator dimension and/or its refractive index parameters by 
+the quantity comparable with their original values. One approach to 
+solve this problem consists in using Fabry-Perot microresonators 
+with tunable mirror separation which contain the optical materials 
+under interest [12, 21, 22]. Additional flexibility of tuning can be 
+achieved by employing Fabry-Perot microresonators with a liquid 
+material inside [21] or translating a wedge-shaped solid optical 
+material to vary its dimensions inside the Fabry-Perot 
+microresonator [23].  
+ 
+Fig.	1.  (a) Coplanar bent optical fibers touching each other. The fiber 
+profile is manipulated by bending and translation of the fiber tails 
+indicated by curved and straight arrows. (b) Illustration of coupling 
+between the input-output microfiber and WGMs in Fiber 1 and Fiber 2 
+near cutoff wavelengths.  
+ Alternatively, of special interest is attaining the eigenwavelength 
+separation tunability in three-dimensional monolithic high Q-factor 
+microresonators, e.g., those with spherical, toroidal, and bottle 
+
+(a)
+Fiber 2
+Fiber 1
+(b)
+Coupled
+WGMs
+(12)
+Fiber 2
+(z)
+2n2
+nin?
+Direct
+Fiber 1
+l1n1
+contact
+x
+Microfiber
+Kn1
+Zshapes. This will allow us to add tunability to the emerging 
+applications of these microresonators in QED, optomechanics, 
+lasing, and frequency comb generation noted above. However, the 
+deformation of most of these monolithic microresonators to achieve 
+significant change of their eigenwavelength separation is unfeasible. 
+ A unique exception, though, is exhibited by SNAP (Surface 
+Nanoscale Axial Photonics) microresonators [24]. These 
+microresonators are introduced at the surface of an optical fiber by 
+its nanoscale deformation, which causes the nanoscale variation of 
+the cutoff wavelengths (CWs) controlling the slow propagation of 
+whispering gallery modes (WGMs) along the fiber axis (see [24, 25] 
+and references therein). In Ref. [26], a SNAP microresonator 
+induced and fully reconfigurable by local heating of an optical fiber 
+was demonstrated. In Ref. [27], it was shown that it is possible to 
+create a SNAP microresonator and control its dimensions by local 
+bending of an optical fiber. Both approaches allow for tuning of 
+eigenwavelength separation of microresonators by the quantity   
+comparable to or larger than its original value. However, in both 
+approaches, the induced microresonator shapes had limited 
+flexibility and their characteristic axial dimensions could not be 
+reduced below several millimeters. In the first case, this restriction 
+was caused by the imposed length of the characteristic heat 
+distribution along the fiber. In the second case, the reduction of 
+microresonator size was limited by the smallest curvature radius 
+corresponding to the fiber breakage threshold.  
+ In this paper we report on our discovery of a new type of WGM 
+optical microresonators which belongs to the group of SNAP 
+microresonators. We show that side coupled coplanar bent fibers 
+(Fig. 1) can induce a high Q-factor SNAP microresonator localized 
+in the region of fiber coupling. The configuration of fibers shown in 
+Fig. 1 allows us to flexibly tune the shape of the induced SNAP 
+microresonators and their axial dimensions from several tens of 
+microns to several millimeters and, respectively, tune their 
+eigenwavelength separation from hundreds of picometers to a 
+picometer. 
+2. Cutoff wavelengths of uncoupled and side-
+coupled straight fibers 
+First, it is instructive to consider the behavior of CWs for 
+uncoupled and side-coupled straight optical fibers. For this 
+purpose, we cleave a 125-micron diameter uncoated 
+commercial optical fiber into two pieces (Fiber 1 and Fiber 2), 
+which are then coaxially aligned and put into contact along 
+3.5 mm of their length as shown in Fig. 2(a). Light is launched 
+into Fiber 1 by a transversely oriented taper with the 
+micrometer 
+diameter 
+waist 
+(input-output 
+microfiber) 
+connected to the Optical Spectrum Analyzer (OSA). After 
+coupling into Fiber 1, light forms WGMs propagating along 
+the fiber surface. In the region of direct contact of fibers (Fig. 
+2(a)), WGMs in Fiber 1 and Fiber 2 are coupled to each other. 
+ To characterize the effect of interfiber coupling, we measured the 
+spectrograms of the configured fiber system. For this purpose, the 
+input-output microfiber was translated along Fiber 1 (Figs. 1(b) and 
+2(a)) touching it periodically with the spatial resolution of 2 µm. At 
+the cut end of Fiber 1, the microfiber was moved towards Fiber 2 and 
+continued scanning Fiber 2. The spectrograms of transmission 
+power 𝑃�𝜆, 𝑧� were measured as a function of wavelength  and 
+microfiber position z along the axis of Fiber 1.  
+ 
+Fig.	2. (a) Illustration of side-coupled straight optical fiber configuration. 
+(b) Spectrogram of this configuration. (c) Magnified section outlined in 
+the spectrogram (b).  
+ The measured spectrogram of our fiber system is shown in Fig. 
+2(b). The left- and right-hand sides of this spectrogram show the 
+spectrograms of uncoupled Fiber 1 and Fiber 2, respectively. Lines 
+in spectrogram shown in Fig. 2(b) indicate the CWs of uncoupled 
+and coupled fibers. These CWs correspond to WGMs with different 
+azimuthal and radial quantum numbers. The magnified copy of the 
+section outlined in Fig. 2(b) is shown in Fig. 2(c). It is seen that the 
+CWs appear as straight lines slightly tilted with respect to the 
+horizontal direction. From the measured magnitude of tilt,  ε� �
+0.015 nm/mm, we determine the linear variation of the fiber radius 
+∆𝑟� � 𝑟�ε�/𝜆� � 0.6 nm/mm [28]. In the latter rescaling relation, 
+we used 𝑟� � 62.5 µm and 𝜆� � 1.55 nm. By linear extrapolation 
+of CWs of Fiber 1 and Fiber 2 (dashed white lines), we confirm that, 
+as expected, their positions (horizontal black dashed line) coincide at 
+the cut ends of these fibers.    
+ At the 3.5 mm long region of fiber touching, WGMs in Fiber 1 
+couple to WGMs in Fiber 2 and the corresponding CWs split. The 
+structure and positions of CWs in the touching region depend on the 
+magnitude of coupling and will be further discussed in Section 4. 
+Here we note that the value of CW splitting found, e.g., from Fig. 
+2(c) is ~ 0.1 nm, which coincides with characteristic values of CW 
+variation in SNAP microresonators [24, 25]. In particular, the 
+positive CW shift in the coupling region leads to the WGM 
+localization and creation of a microresonator which can be tuned by 
+changing the length of the side-coupled fiber segment. In our current 
+experiment, the Q-factor of the induced SNAP resonator was poor 
+due to the scattering of light at the imperfectly cleaved fiber ends, 
+
+(a)
+Nottoscale
+3.5 mm
+Fiber2
+Fiber1
+125μm
+Microfiber
+1549.00
+(b)
+1548.50
+-5
+TransmissionPower (dB)
+(wu)
+1548.00
+length
+1547.50
+10
+Wavel
+1547.00
+15
+1546.50
+1546.00
+-20
+1545.50
+0
+1
+2
+3
+4
+5
+6
+Distance along fiber (mm)
+0
+1548.50
+(c)
+1548.40
+-5
+length
+1548.20
+-10
+1548.10
+Javel
+15
+1548.00
+1547.90
+-20
+1547.80
+0
+1
+2
+3
+4
+5
+6
+Distancealongfiber(mm)which, typically, ensure around 70% WGM reflectivity [29]. 
+Nevertheless, we suggest that the demonstrated resonator can be 
+directly used to create miniature broadly tunable optical delay lines 
+generalizing our previous results based on the SNAP 
+microresonators with fixed dimensions [30, 31]. Indeed, in these 
+devices the WGM pulses complete only a single round trip along the 
+fiber axis and therefore their attenuation at the fiber facets may 
+reduce the output light power by around 50% only. We also suggest 
+that, after feasible improvement, the Q-factor of these 
+microresonators can be significantly improved as further discussed 
+in Section 6. 
+3. Basic experiment  
+In our proof-of-concept experiments, we used 125-micron diameter 
+uncoated commercial silica optical fibers touching each other as 
+shown in Fig. 1(a). The ends of Fiber 1 and Fiber 2 were bent and 
+translated to arrive at the required profile of these fibers near their 
+coupling region illustrated in Fig. 1(b). The fibers used were either 
+originally straight or preliminary softened in a flame and bent 
+permanently. As described in the previous section, WGMs were 
+launched into Fiber 1 by a transversely oriented microfiber 
+connected to the OSA. If the separation between Fiber 1 and Fiber 2 
+is small enough, WGMs penetrate from Fiber 1 into Fiber 2.  
+ In the simplest configuration considered in this Section, Fiber 1 
+was straight, and coplanar Fiber 2 was bent. The fibers were put in 
+contact and then slightly pushed towards each other to increase the 
+coupling region. The photograph of the fiber configuration used in 
+this experiment is shown in Fig. 3(a). From this picture, we estimated 
+the curvature radius of the bent fiber as 𝑅~30 mm (see further 
+discussion of the fiber profile in Section 4). Fig. 3(b) shows the 
+spectrogram of the configured structure measured along the 3.5 nm 
+bandwidth within the 700 µm axial length of Fiber 1. At the edges of 
+the scanned region, the interfiber coupling is negligible. In these 
+regions, CWs do not noticeably change with distance 𝑧 and, thus, 
+correspond to Fiber 1 only. The arrangement of CWs in these regions 
+is similar to that in Fig. 2(b).
+ 
+Fig.	3.  (a) Photograph of the side-coupled fibers used in the experiment. The upper fiber is bent with the curvature radius 𝑅~30 mm and the lower 
+fiber has the curvature radius greater than 1 m. (b) The spectrogram measured in the vicinity of the coupling region of these fibers. (b1) and (b2) 
+Spectrograms showing the magnified sections outlined in the spectrogram (b). (c1) and (c2) Spectrograms of the microresonators numerically 
+calculated in the two-mode approximation detailed in the text, which replicate the experimental spectrograms in Figs. (b1) and (b2), respectively.    
+
+(a)
+R~30 mm
+Experiment
+3-10
+nsertion
+1mm
+-15
+(b1)
+1549.65
+1549.46
+1549.48
+1547.45
+(b)
+(b2)
+Wavelength (um)
+1549.60
+1550.00
+-2
+1547.40
+1549.55
+(b2)
+-4
+1549.50
+1549.50
+-6
+Transmission Power (dB)
+1549.45
+Wavelength (nm)
+1549.00
+-8
+1549.40
+1547.20
+1549.35
+1548.50
+-10
+1547.15
+1549.30
+-12
+1548.00
+200
+400
+600
+200
+400
+600
+-14
+Distancealongfiber(um)
+Distance along fiber (μm)
+1547.50
+(b1)
+-16
+-18
+1547.00
+Theory
+-20
+0
+200
+400
+600
+Distance along fiber (μm)
+(c1)
+(c2)0.4
+0.4
+-5
+Wavelengthvariation(nm)
+Wavelength variation (nm)
+0.3 
+0.3
+0.2
+ 0.2
+-15
+0.1
+0.1
+0.0
+0.0
+-20
+0
+200
+400
+600
+0
+200
+400
+600
+Distancealongfiber(um)
+Distance along fiber (μm) The effect of coupling shows up in the central region of the 
+spectrogram in Fig. 3(b). In this region, different CWs exhibit 
+different positive and negative variations along the axial length 𝑧. 
+The exemplary regions of this spectrogram named (b1) and (b2) are 
+magnified in Figs. 3(b1) and 3(b2), respectively. It is seen that, as 
+expected, in contrast to negative variations, positive CW variations 
+lead to the WGM confinement and the creation of microresonators. 
+Our estimates illustrated in the inset of Fig. 3(b2) show that the Q-
+factor of the created microresonator (which measurement was 
+limited by the 1.3 pm resolution of the OSA used) exceeds 10�. The 
+observed CW variations in Figs. 3(b1) and (b2) can be explained by 
+the theory described below. 
+4. Basic theory 
+We assume that the fiber bending is small enough so that the 
+propagation of light along the axial direction of side-coupled 
+fibers (Fig. 1(b)) can be considered as propagation along a 
+single waveguide with asymmetric cross-section including 
+both fibers. The wavelengths of slow WGMs are close to the 
+CWs 𝜆��𝑧� of this compound waveguide. To determine the 
+complex-valued CWs 𝜆��𝑧�, we introduce the original CWs 
+𝜆��� � �
+�𝛾��� and 𝜆��� � �
+�𝛾��� of unbent Fiber 1 and Fiber 2 
+with the imaginary parts determined primarily by material 
+losses and scattering of light at the fiber surface. We assume 
+that there are 𝑁� and 𝑁� cutoff wavelengths in Fibers 1 and 
+Fiber 2, respectively, which contribute to the resonant 
+transmission, so that 𝑛� � 1,2, … , 𝑁�, 𝑗 � 1,2. We refer to the 
+integers 𝑛, 𝑛� and 𝑛� as to the transverse quantum numbers. 
+Variation of 𝜆��𝑧� is caused by bending of fibers [27] and, in 
+our case, primarily by their coupling. In the absence of the 
+input-output fiber, the CWs of our system, 𝜆 � 𝜆��𝑧�, 𝑛 �
+1,2, … , 𝑁� � 𝑁�,  are determined as the roots of the 
+determinant: 
+ 
+
+
+det
+( )
+0
+z
+ 
+
+I
+Ξ
+         (1) 
+ 
+Here 𝐈  is the unitary �𝑁� � 𝑁�� � �𝑁� � 𝑁��  matrix and 
+matrix 
+ 
+1
+1
+12
+†
+12
+2
+2
+( )
+( )
+( )
+( )
+( )
+z
+z
+z
+z
+z
+
+
+
+ 
+
+
+
+
+Λ
+Δ
+Δ
+Ξ
+Δ
+Λ
+Δ
+.      (2) 
+ 
+The submatrices in Eq. (2) determine the original CWs of 
+Fiber 1 and Fiber 2, 𝚲� � �𝜆��� � �
+�𝛾����, couplings inside 
+each of the fiber caused by bending, 𝚫��𝑧� � �δ����
+���
+�𝑧��, and 
+interfiber 
+couplings 
+𝚫���𝑧� � �δ����
+���� �𝑧��
+, 
+𝑚�, 𝑛� �
+1,2, … 𝑁�.  
+ As in SNAP [24], dramatically small nanometer and sub-
+nanometer scale variations of CWs 𝜆��𝑧� along the compound 
+fiber waveguide can localize WGMs and induce an optical 
+microresonator having eigenwavelengths 𝜆��  with axial 
+quantum numbers 𝑞. Due to the smooth and small CW variation 
+and proximity of the localized WGM wavelengths 𝜆��  to 
+𝜆��𝑧�, the corresponding eigenmode can be presented as 
+𝐸���𝑥, 𝑦, 𝑧� � Ψ���𝑧�Ω��𝑥, 𝑦, 𝑧� where the transverse WGM 
+distribution Ω��𝑥, 𝑦, 𝑧�  is calculated at the CW 𝜆��𝑧� and 
+depends on 𝑧 parametrically slow [32], and function Ψ���𝑧� 
+determines the axial dependence of the microresonator 
+eigenmode amplitude and satisfies the one-dimensional wave 
+equation [24] 
+ 
+2
+3/2
+2
+2
+3/ 2
+2
+( , )
+0,
+( , )
+( )
+.
+n
+r
+n
+n
+n
+n
+n
+d
+n
+z
+z
+z
+dz
+
+
+
+
+
+
+
+
+ 
+ 
+
+
+   (3) 
+ 
+where 𝑛� is the refractive index of the fibers. 
+ The coupling parameters 𝜅���𝑧�between WGM 𝐸���𝑥, 𝑦, 𝑧� 
+and the input-output wave in the microfiber is determined by 
+their overlap integral. Commonly, the microfiber diameter is 
+much smaller than the characteristic axial variation length of 
+𝐸���𝑥, 𝑦, 𝑧�.  For this reason, similar to the analogous 
+approximation in the SNAP platform [24, 33], the coupling 
+parameters 𝜅���𝑧�  are proportional to the values of 
+𝐸���𝑥, 𝑦, 𝑧�  at the axial coordinate 𝑧 of the input-output 
+microfiber. Then, calculations based on the Mahaux-
+Weidenmüller theory [34-36] presented in Supplementary 
+Material allowed us to express the transmission power 𝑃�𝜆, 𝑧� 
+through the input-output microfiber coupled to the considered 
+fiber configuration (Fig. 1(b)) as  
+ 
+1
+2
+1
+2
+2
+*
+1
+1
+1
+( )
+( , , )
+( , )
+1
+( )
+( , , )
+N
+N
+n
+n
+n
+N
+N
+n
+n
+n
+D z G z z
+P z
+D z G z z
+
+
+
+
+
+
+
+
+
+
+
+
+.      (4) 
+ 
+Here 𝐺��𝑧�, 𝑧�, 𝜆) is the Green’s function of Eq. (3). Eq. (4) 
+generalizes the expression for the transmission power 
+previously derived in Ref. [24]. As shown below, functions 
+𝐷��𝑧� can be expressed through and have characteristic values 
+similar 
+to 
+the 
+coupling 
+D-parameters 
+which 
+were 
+experimentally measured previously and typically have the real 
+and imaginary parts ~ 0.01 µm-1 [24, 33]. Close to the resonance 
+condition, 𝜆 � 𝜆��, for sufficiently small losses and coupling, and 
+separated CWs 𝜆��𝑧�, only one Green’s function with number 𝑛 
+contributes to the sums in Eq. (4). Then, Eq. (4) coincides with that 
+previously derived in Ref. [24]. However, generally, the 
+contribution of more than one term to the sums in Eq. (4) may be 
+significant.    
+Before the detailed description of the spectrograms in Figs. 
+2(b) and 3(b), we note that the transmission power plots in 
+these figures characterize the CWs of the coupled fiber system 
+determined by Eq. (1) viewed by the input-output microfiber 
+and, subsequently, OSA. Therefore, the CWs of Fiber 2, which are 
+the solutions of Eq. (1) but uncoupled from Fiber 1 cannot be 
+seen by the OSA. On the other hand, the number of CWs which 
+can show up in the coupling region can be as many as 𝑁� � 𝑁�, 
+i.e., significantly greater than the number ��� of visible 
+uncoupled CWs of Fiber 1 (see Fig. 2(b) as an example).  
+ To clarify the effect of coupling between WGMs in adjacent 
+fibers, we consider the two-mode approximation, 𝑁� � 𝑁� � 1, 
+assuming that the wavelength 𝜆 of the input light is close to an 
+
+unperturbed single WGM CW 𝜆�� � �
+�𝛾 of Fiber 1 and a single CW 
+𝜆�� � �
+�𝛾 of Fiber 2 having the same imaginary part. Consequently, 
+in Fig. 1(b) we now set 𝑛� � 𝑛� � 1. We neglect the effect of the 
+CW variation due to the fiber bending [27], which is usually smaller 
+than the effect of fiber coupling, setting 𝛿��
+��� � 0. Then, the CWs 
+𝜆��𝑧� and 𝜆��𝑧� of the compound fiber are found from Eq. (1) as 
+ 
+
+
+
+
+
+
+2
+2
+(12)
+1,2
+11
+21
+11
+21
+11
+1
+1
+( )
+( )
+2
+4
+z
+i
+z
+
+
+
+
+
+
+
+
+
+
+
+
+
+  (5) 
+ 
+ The dependence on the transverse coordinates 𝑥 and 𝑦 (Fig. 1(b)) 
+of the compound WGM corresponding to CWs 𝜆��𝑧�  can be 
+calculated as follows. We introduce the unperturbed WGMs in Fiber 
+1 and 2 (considered unbent and uncoupled) calculated at their CWs 
+𝜆�� and 𝜆�� as  Ω�
+����𝑥, 𝑦� and   Ω�
+����𝑥, 𝑦�. Then, in the two-mode 
+approximation, the compound modes generated by weak coupling 
+of modes Ω�
+����𝑥, 𝑦� and   Ω�
+����𝑥, 𝑦� are determined as [37] 
+  
+(1)
+(2)
+1
+1
+1
+2
+2
+(1)
+(2)
+2
+1
+1
+2
+2
+(12)
+11
+11
+21
+1
+( )
+( , , )
+( , )
+( , ),
+1
+( )
+1
+( )
+( )
+1
+( , , )
+( , )
+( , ),
+1
+( )
+1
+( )
+( )
+                                ( )
+.
+z
+x y z
+x y
+x y
+z
+z
+z
+x y z
+x y
+x y
+z
+z
+z
+z
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+ 
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+  (6) 
+ 
+Consequently, the coupling parameters to the microfiber entering 
+Eq. (4) at coordinate 𝑧 are 
+ 
+1
+2
+2
+2
+( )
+( )
+,
+( )
+,
+1
+( )
+1
+( )
+D
+D
+z
+D z
+D z
+z
+z
+
+
+
+
+ 
+
+
+
+
+
+   (7) 
+ 
+where 𝐷 is the z-independent coupling parameter between the input-
+output microfiber and Fiber 1 [24, 33].  
+ To map the bent fiber axial profile ℎ�𝑧� to the CW envelope 
+profiles of the induced microresonators, we have to determine the 
+relation between ℎ�𝑧� and coupling coefficient  𝛿��
+�����𝑧�. Similar 
+to calculations in Refs. [38, 39], for the smooth and small ℎ�𝑧� 
+considered here, we find 
+ 
+
+
+1/2
+(12)
+2
+11
+0
+2
+( )
+exp
+1
+( )
+r
+z
+n
+h z
+
+
+
+
+
+
+
+
+
+
+
+
+
+,     (8) 
+ 
+where 𝛿� is 𝑧-independent. Assuming the simplest profile of the 
+bent fiber having the curvature radius 𝑅 as 
+ 
+ℎ�𝑧� � 𝑧�/2𝑅         (9)  
+ 
+for silica fibers with 𝑛� �1.44, we estimate the FWHM of 𝛿��
+�����𝑧� 
+as 𝑧����~0.5�𝜆𝑅��/�. At 𝜆~1.55 µm and 𝑅~30 mm of our 
+experiment, we have 𝑧����~100 µm. From Eqs. (5) and (8), we 
+find that the FWHM of the CW, depending on the value of 𝜆�� �
+𝜆��, is between 𝑧���� and 2𝑧���� which is only in qualitative 
+agreement with the microresonator FWHM 𝑧����~ 250  µm 
+found from experimental data in Figs. 3(b1) and (b2).  
+ The results of our numerical modeling in the two-mode 
+approximation considered based on Eqs. (3)-(9) are shown in Figs. 
+3(c1) and 3(c2). To fit the experimental data, we set the average CW 
+0.5(𝜆�� � 𝜆��) = 1.55 µm, the CW difference 𝜆�� � 𝜆�� � 0.05 
+nm in Fig. 3(c1) and 𝜆�� � 𝜆�� � �0.05  nm in Fig. 3(c2), 
+coupling parameter 𝐷 � �0.01 � 0.01𝑖 µm-1 [24, 33], Q-factor 
+𝑄 � 10�, the microresonator FWHM 𝑧����~ 250 µm and its 
+spectral height  ~ 0.15 nm, similar to these values found from Figs. 
+2(b1) and (b2).  
+ The experimental spectrograms in Fig. 3(b1) and (b2) and 
+theoretical spectrograms in Figs. 3(c1) and (c2) look nicely similar. 
+However, important differences between them should be noted. 
+From Eqs. (8) and (9), the FWHM value 𝑧����~ 250  µm 
+corresponds to the Fiber 2 curvature radius 𝑅~66 mm, which is 
+twice as large as that measured from the fiber image shown in Fig. 
+3(a). We suggest that the difference is caused by the deviation of the 
+shape of Fiber 2 from parabolic in the coupling region as well as by 
+the fiber misalignment. The additional deformation of fibers may be 
+induced by their electrostatic attraction and pressuring, which are not 
+visible in Fig. 3(a). Our suggestion is confirmed by the experimental 
+profiles of the induced microresonator envelopes and CW shapes in 
+Figs. 3(b1) and (b2) which, as compared to those in the theoretical 
+spectrograms in Figs. 3(c1) and (c2), have larger side slopes and are 
+flatter in the middle. Next, we notice that, in the theoretical 
+spectrograms, the CW wavelength profiles are more mirror-
+symmetric to the microresonator envelopes with respect to the 
+horizontal line (following Eq. (5)), while, in the experimental 
+spectrograms, the lower CW profiles are shallower than the 
+microresonator envelopes. We suggest that this deviation can be 
+eliminated by taking into account the coupling with other WGMs 
+ignored in the two-mode approximation considered.  
+5.  Tunability 
+Bending and translating the tails of Fiber 1 and Fiber 2 side-
+coupled to each other as illustrated in Fig. 1 allowed us to tune 
+the dimensions of the fiber coupling region and thereby tune the 
+dimensions of created microresonators. As in the previous 
+sections, in our experiments we used 125 µm optical fibers. We 
+investigated the cases of the smallest microresonators 
+containing a few wavelength eigenvalues and having the 
+characteristic axial dimensions of hundred microns (Figs. 5(a1)-
+(a4)), as well as larger microresonators with dimensions of 
+several hundred microns (Figs. 5(b1)-(b4) and (c1)-(c3)) and 
+the largest microresonator having the axial length of 5 
+millimeters (Fig. 4(d)). 
+Considering the smallest microresonators, we monitored the 
+process of their creation. Side-coupling of a straight Fiber 1 and 
+Fiber 2 bent with a sufficiently small curvature radius of ~ 1 mm 
+introduced small perturbation in CWs shown in the 
+spectrogram in Fig. 4(a1). Increasing the fiber radius further, we 
+arrived at the microresonator with a single eigenwavelength 
+(Fig. 4(a2)). The inset inside the Fig. 4(a2) spectrogram, which 
+magnifies the region near this eigenwavelength, shows that the 
+axial dimension of the corresponding eigenmode is ~ 200 µm. 
+Remarkably, except for the axial dimension of localized WGMs 
+with 
+uniform 
+magnitude 
+in 
+specially 
+designed 
+bat 
+
+microresonators [39, 40], this dimension (which expansion is 
+critical, e.g., for QED applications [41]) is the record large 
+characteristic 
+WGM 
+dimension 
+demonstrated 
+in 
+microresonators to date. The measured Q-factor of this 
+microresonator (limited by the 1.3 pm resolution of the OSA 
+used) was slightly greater than 10�.  
+ Larger bending radii of Fiber 2 having the order of 10 mm led to 
+the creation of microresonators with millimeter-order axial 
+dimensions having the spectrograms shown in Figs. 5(b1)-(b4) and 
+(c1)-(c3). The close to parabolic shape of these microresonators 
+suggests that they can be used, e.g., as tunable optical frequency 
+comb generators [42]. We note that the behavior of the CWs and 
+microresonators envelopes in most of these spectrograms cannot be 
+accurately described by the two-mode approximations of Section 4. 
+Of particular interest is the spectrogram shown in Fig. 4(c2). At first 
+sight, the envelop of the microresonator in this spectrogram is the 
+continuation of the CW of Fiber 1 (compare with Figs. 3(b1) and 
+(c1)). Unexpectedly, the axial WGM localization in this 
+microresonator (caused by the WGM reflection from the CW-
+generated turning points [24]) sharply dissolves inside the 
+microresonator area. 
+ 
+Fig.	4.	Tunability of microresonators. (a1)-(a4) Spectrograms of induced microresonators for small curvature radius of Fiber 2 ~ 1 mm.  (b1)-(b4) and 
+(c1)-(c3) spectrograms of induced microresonators for a lager radius of Fiber 2 ~ 10 mm. (d) Spectrogram of a 5 mm long microresonator induced by 
+touching straight Fiber 1 and Fiber 2 which was preliminary permanently bent  at the ends as shown in the inset.  
+
+(a1)
+(a2)
+(a3)
+(a4)
+R=1.2mm
+R=1.5mm
+R=1.6mm
+1.7mm
+1546.80
+1546.80
+1546.80
+1546.80
+ap
+(w
+(wu
+1546.6
+(wu
+(wu
+1546.70
+1546.70
+-5
+-45
+-5
+1546.60
+1546.60
+-6
+1546.60
+1546.60
+0200400600
+0200400600
+0200400600
+0200400600
+Distancealongfiber(μm)
+Distancealongfiber(um)
+Distancealongfiber(μm)
+Distancealongfiber(um)
+(b1)
+(b2)
+(b3)
+(b4)
+R=6.1mm
+R=7.7mm
+R=8.1mm
+R=16.3mm
+1551.90
+1551.90
+1551.90
+1551.90
+10
+10
+1551.80
+-20
+1551.80
+32
+1551.70
+1551.70
+1551.70
+1551.70
+lavele
+4
+-5
+1551.60
+1551.60
+≤1551.60
+1551.60
+-5
+1551.50
+1551.50
+1551.50
+200400600800
+200400600800
+200400600800
+1551.50
+0
+200400600800
+Distancealongfiber(um)
+Distancealongfibor(μm)
+Distancealongfiber(um)
+Distancealong fiber(um)
+(c1)
+(c2)
+(c3)
+R=18mm
+R~30mm
+R~30mm
+1551.80
+1551.80
+Transmission Power (dB)
+1551.80
+2
+(dB)
+-2
+1551.70
+4
+-4
+nbu
+.6
+6
+1551.60
+1551.60
+AeM
+-8
+8
+1551.50
+1551.50
+1551.50
+-10
+-10
+-10
+0
+400
+800
+1200
+1600
+400
+800
+1200
+1600
+0
+400
+800
+1200
+1600
+Distancealong fiber(um)
+Distancealongfiber(um)
+Distancealongfiber (um)
+(d)
+1548.60
+length
+4
+6
+-10
+1548.30
+0
+1000
+2000
+3000
+4000
+5000
+6000
+Distancealong fiber(um)To create longer microresonators, we, first, permanently bent 
+the tails of Fiber 2 as illustrated in the inset of Fig. 4(d). This 
+allowed us to arrive at an arbitrarily large curvature radius of 
+this fiber including its straight shape between the bent tails. As 
+an example, Fig. 4(d) shows the spectrogram of a 5 mm long 
+microresonator. Though the eigenwavelength width of this 
+microresonator is greater than its free spectral range, we 
+suggest that, in contrast to the lossy microresonators induced by 
+side-coupled cleaved straight fibers demonstrated in Section 2, 
+its Q-factor is similar to that of the smaller microresonators 
+considered in this section and Section 3.   
+6. Discussion 
+The effect of induction of high Q-factor WGM tunable optical 
+microresonators in side-coupled optical fibers discovered in this 
+paper enables a range of exciting generalizations and applications. 
+Further extension of tuning flexibility can be achieved by enabling 
+different boundary conditions at the fiber tails (Fig. 1(a)), different 
+interfiber touching stresses, and different preliminary permanent 
+fiber bending.  
+ Configurations of fibers, which are potentially attractive for future 
+research and applications, are illustrated in Fig. 5. Fig. 5(a) shows a 
+way to create long microresonators alternative to the method 
+utilizing fibers with permanently bent tails illustrated in Fig. 4(d). In 
+the configuration of Fig. 5(a), the length of the induced 
+microresonator increases as the curvature radii of touching fibers 
+approach each other. Provided that the variation of the fiber radii can 
+be performed so that the parabolicity of the induced microresonators 
+was maintained, the configuration of Fig. 5(a) can serve for the 
+generation of the optical frequency combs with a tunable repetition 
+rate.  
+ 
+Fig.	5.	(a) Bent fibers with increased coupling region. (b) Bent fibers 
+with increased coupling region and abrupt side of the induced 
+microresonator. (c) A bottle microresonator side-coupled to a fiber. (d) 
+Side coupled straight fibers with tapered facets forming a rectangular 
+microresonator.  (e) Two straight fibers with tapered facets coupled to 
+the third straight fiber forming a rectangular microresonator. (f) 
+Twisted side-coupled fibers. (g) A microcapillary fiber filled with liquid 
+and side coupled to a bent fiber. (h) Three straight coupled fibers.   
+ In Fig. 5(b), the lower fiber is terminated with a short taper, which 
+can be introduced using, e.g., a CO2 laser. Simple estimates show 
+that a taper with a characteristic length of 100 µm at the end of a 125 
+µm diameter optical fiber creates an abrupt CW barrier with a slope 
+of ~ 100 nm/µm at 1.5 µm wavelength. The steepness of the slope 
+of this barrier (critical for impedance matching of light from the 
+input-output microfiber [43]) is 100 times greater than that 
+demonstrated in Ref. [44] with the femtosecond laser inscription. 
+The configuration shown in Fig. 5(b) can be used for the creation of 
+miniature dispersionless tunable optical delay lines provided that the 
+shape of the induced microresonator is kept semi-parabolic in the 
+process of tuning [43].  
+ Experimental investigation and development of the theory of 
+WGMs in a microresonator side-coupled to an optical fiber is of 
+particular interest. Fig. 5(c) illustrates the side coupling of a fiber and 
+a bottle microresonator. While the fiber is open-ended, coupling of 
+the bottle microresonator to the straight fiber can cause the 
+localization of light in the fiber, similar to the coupling between bent 
+optical fibers considered above. The configuration shown in Fig. 5(c) 
+suggests a way of tuning the microresonator eigenwavelengths.  
+ The fiber configuration shown in Fig. 5(d) is similar to two 
+straight side-coupled fibers considered in Section 2. To improve the 
+Q-factor of the microresonator induced along the coupling region, 
+the cleaved ends of fibers shown in Fig. 2(a) are modified by the 
+tapered ends. The configuration of fibers shown in Fig. 5(e) 
+illustrates an alternative way to create tunable microresonators when 
+the position of both their sides can be tuned. The rectangular 
+microresonators induced in both configurations can be used for the 
+creation of tunable delay lines which, as shown in Ref. [31], can be 
+dispersionless with a good accuracy. 
+ The coupling of twisted optical fibers illustrated in Fig. 5(f) is 
+interesting to investigate both theoretically and experimentally. In 
+the cylindrical coordinates �𝑧, 𝜌, 𝜑� of one of the fibers, the curve 
+along which the fibers touch each other corresponds to the azimuthal 
+angle 𝜑 � 𝜑� � 𝛼𝑧 , where 𝛼  is the twisting coefficient. The 
+corresponding value of the WGM field is proportional to 
+exp �𝑖𝛽𝑧 � 𝑖𝑚�𝜑� � 𝛼𝑧�� where 𝛽 is the propagation constant. 
+From this expression, a WGM at CW corresponding to 𝛽 � 0 is 
+seen by another fiber as a mode with nonzero propagation constant. 
+Thus, in contrast to the untwisted fibers, coupling between the side-
+coupled twisted fibers is essentially three dimensional. 
+ Fig. 5(g) shows a microcapillary fiber filled with liquid and side-
+coupled to a bent fiber. For the microcapillary with sufficiently thin 
+walls, a microresonator induced inside it by the side-coupled fiber 
+performs nonlocal sensing of liquid [45]. In Refs. [46] and [47], such 
+microresonators were introduced with the CO2 laser and slow 
+cooking methods. Fig. 5(g) suggests the simplest approach for the 
+realization of nonlocal microfluidic sensing.  
+   Fig. 5(h) illustrates three straight side-coupled fibers. In contrast 
+to two coupled fibers, WGMs launched into this configuration will 
+propagate into both azimuthal direction and, in particular, into the 
+positive and negative directions of the input-output microfiber with 
+approximately the same amplitudes. The channel formed between 
+these fibers can be used for gas and microfluidic sensing. Unlike the 
+microcapillary illustrated in Fig. 5(g), no ultrathin wall enabling the 
+WGM sensing of the internal channel is required in this case.   
+While the model of two coupled CWs developed here 
+qualitatively explains some characteristic features of the 
+
+(a)
+(f)
+(b)
+(g)
+(c)
+(d)
+(h)
+(e)experimentally measured spectrograms, the complete explanation 
+and quantitative fitting of the experimental data should include the 
+effect of several CWs and be based on the further development of 
+the coupled wave theory. The future theory should also allow us 
+to express the fiber profiles and deformation in the region of 
+coupling through the values of forces and moments applied to 
+the fiber tails (Fig.1(a)) including the effect of electrostatic fiber 
+attraction.  
+We suggest that the fixed submicron-wide gaps between 
+coupled fibers and input-output microfiber, rather than their 
+direct contact considered here, will allow us to demonstrate the 
+proposed microresonators with the Q-factor exceeding 108 [8]. 
+While such large Q-factors are not required for the realization of 
+tunable delay lines [43], signal processors [25], and microlasers 
+[19-21], they may be important for the realization of frequency 
+comb generators with tunable repetition rate [15, 16, 42], as 
+well as for the cavity QED [8, 11,12] and optomechanical 
+applications [13, 14].  
+ 
+ 
+ 
+Supplementary material 
+Expression for the transmission power    
+We introduce the discrete eigenwavelengths of the microresonator 
+in the compound fiber system, 𝜆� � �
+�𝛾� , 𝑚 � 1,2, … , 𝑀 and 
+coupling 
+coefficients 𝜅��𝑧�  between 
+the 
+corresponding 
+eigenmodes and the input-output microfiber positioned at axial 
+coordinate 𝑧. We calculate the transmission power 𝑃�𝜆, 𝑧� of our 
+system by applying the Mahaux-Weidenmüller formula [34-36]: 
+ 
+
+
+2
+1
+†
+†
+2
+( , )
+1
+( , ) ,
+( , )
+( )
+( )
+( )
+( )
+( )
+i
+P
+z
+iT
+z
+T
+z
+z
+z
+z
+z
+
+
+
+
+
+
+
+
+
+Κ
+Δ
+Κ
+Κ
+Κ
+,    (S1) 
+ 
+where 
+  
+1
+2
+†
+2
+1
+1
+2
+1
+1
+2
+2
+( )
+( )
+( )
+( )
+( )
+( ) ,
+( )
+,
+...
+( )
+0
+...
+0
+0
+...
+0
+( )
+.
+...
+...
+...
+...
+0
+0
+...
+i
+M
+i
+i
+i
+M
+M
+z
+z
+z
+z
+z
+z
+z
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+ 
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+ 
+
+
+
+
+
+
+
+
+
+Θ
+Δ
+Κ
+Κ
+Κ
+Δ
+ (S2)  
+ 
+It is assumed in Eq. (S1) that the coupling to the input-output 
+waveguide 
+does 
+not 
+introduce 
+the 
+shifts 
+of 
+the 
+eigenwavelengths [35] which will be added later. We simplify 
+the expression for the transmission power by expanding the 
+inverse matrix in Eq. (S1) as follows: 
+ 
+
+
+  
+
+ 
+ 
+
+
+ 
+1
+†
+2
+1
+†
+1
+2
+0
+1
+†
+2
+2
+1
+†
+1
+†
+2
+1
+1
+†
+1
+†
+1
+2
+2
+1
+†
+2
+2
+2
+( )
+( )
+( )
+( )
+( )
+( )
+( )
+1
+( )
+( )
+( )
+( )
+( )
+( ) ( )
+( )
+( )
+...
+( )
+( )
+( ) ( )
+( )
+( )
+...
+( )
+( )
+1
+( )
+( )
+( )
+i
+n
+n
+i
+n
+i
+i
+n
+n
+i
+n
+m
+i
+i
+i
+m
+m
+m
+z
+z
+z
+z
+z
+z
+z
+z
+z
+z
+z
+z
+z
+z
+z
+z
+z
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+Δ
+Κ
+Κ
+Δ
+Κ
+Κ
+Δ
+Δ
+Κ
+Κ
+Δ
+Κ
+Κ
+Δ
+Κ
+Κ
+Δ
+Κ
+Κ
+Δ
+Κ
+Κ
+Δ
+Δ
+Κ
+Κ
+1
+0
+1
+1
+†
+1
+2
+2
+2
+1
+2
+( )
+( )
+( )
+( )
+1
+( )
+( )
+1
+n
+M
+n
+i
+M
+m
+i
+i
+m
+m
+m
+z
+z
+z
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+Δ
+Δ
+Κ
+Κ
+Δ
+  
+ 
+Substituting this expression into Eq. (S1), we find: 
+ 
+2
+2
+2
+1
+2
+2
+2
+1
+2
+( )
+1
+( , )
+( )
+1
+M
+m
+i
+i
+m
+m
+m
+M
+m
+i
+i
+m
+m
+m
+z
+P
+z
+z
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+      (S4) 
+ 
+We separate the series of eigenwavelengths  𝜆� � �
+�𝛾�  and 
+coupling coefficients 𝜅��𝑧� by their correspondence to CWs 𝜆��𝑧� 
+entering Eq. (3) of the main text. For this purpose, we rewrite these 
+parameters as 𝜆�� � �
+�𝛾� and 𝜅���𝑧�, where 𝑞 is the axial quantum 
+number of the eigenmode 𝐸���𝑥, 𝑦, 𝑧� � Ψ���𝑧�Ω��𝑥, 𝑦, 𝑧�.  
+Here Ψ���𝑧� satisfies Eq. (3) and Ω��𝑥, 𝑦, 𝑧� is a parametrically 
+slow function of the axial coordinate  𝑧.  Substituting 𝛾� → 𝛾� we 
+assume that the material losses do not depend on the axial quantum 
+number 𝑞. Then, similar to the arguments of Ref. [24] (see Eq. (13) 
+in this reference), the coupling coefficients can be factorized as 
+�𝜅���𝑧��
+� � 2𝑖𝐷��𝑧��Ψ���𝑧��
+� . Using the expression for the 
+Green’s function of Eq. (3), 
+ 
+2
+2
+( )
+( , , )
+qn
+n
+i
+q
+qn
+n
+z
+G z z 
+
+
+
+
+
+
+
+
+ ,     (S5) 
+ 
+we rewrite Eq. (S4) as 
+ 
+2
+*
+1
+1
+1
+( )
+( , , )
+( , )
+1
+( )
+( , , )
+N
+n
+n
+n
+N
+n
+n
+n
+D z G z z
+P
+z
+D z G z z
+
+
+
+
+
+
+
+
+
+
+.     (S6) 
+ 
+To identify the physical meaning of parameters 𝐷��𝑧�, we recall the 
+expression for the transmission power of a SNAP microresonator 
+under the assumption of a single CW contribution (𝑁 � 1) and 
+
+lossless coupling to the input-output microfiber [24]: 
+ 
+ 
+2
+*
+1
+1
+1
+1
+1
+1
+( , , )
+( , )
+1
+( , , )
+D G z z
+P
+z
+D G z z
+
+
+
+
+
+
+      (S7) 
+ 
+Here complex parameter 𝐷�,  which was experimentally 
+measured and analyzed previously [24, 33], determines the 
+coupling to the input-output microfiber as well as the WGM 
+phase shift due to this coupling. Importantly, while the 
+imaginary part of 𝐷��𝑧�  contributes to the widths of the 
+resonances, its real part (not taken into account in the original Eq. 
+(S1)) determines the WGM phase shifts caused by the coupling to 
+the input-output microfiber.    
+ 
+Funding. The Engineering and Physical Sciences Research Council 
+(EPSRC), grants EP/P006183/1 and EP/W002868/1. Horizon 2020 
+MSCA-ITN-EID grant 814147. 
+Disclosures. The authors declare no conflicts of interest. 
+Data availability. Data underlying the results presented in 
+this paper are not publicly available at this time but may be 
+obtained from the authors upon reasonable request.  
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+ 
+ 
+  
+ 
+

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+Synesthetic Dice: Sensors, Actuators, And Mappings
+Albrecht Kurze
+Chemnitz University of Technology, Albrecht.Kurze@informatik.tu-chemnitz.de
+How bright can you cry? How loud does the sun shine? We developed a multisensory and multimodal tool, the Loaded Dice, for use in 
+co-design workshops to research the design space of IoT usage scenarios. The Loaded Dice incorporate the principle of a technical 
+synesthesia, being able to map any of the included sensors to any of the included actuators. With just a turn of one of the cubical 
+devices it is possible to create a new combination. We discuss the core principles of the Loaded Dice, what sensors and actuators are 
+included, how they relate to human senses, and how we realized a meaningful mapping between sensors and actuators. We further 
+discuss where we see additional potential in the Loaded Dice to support synesthetic exploration – as Synesthetic Dice – so that you can 
+eventually find out who cries brighter.
+CCS CONCEPTS • Human-centered computing~Human computer interaction (HCI)
+Additional  Keywords  and  Phrases:  multisensory,  multimodal,  synesthesia,  design,  ideation,  tools,  methods,  IoT, 
+Internet of Things, haptic technology, cubic shape, tangible interactive devices, input and output devices, tangibles
+ACM Reference Format:
+Albrecht Kurze. 2022. Synesthetic Dice: Sensors, Actuators, And Mappings. In Workshop Sensory Sketching (CHI’22). 
+April 22, 2022. 4 pages.
+1
+INTRODUCTION
+Some years ago we designed and developed the Loaded Dice [8,9], a multisensory and multimodal hybrid toolkit to 
+ideate Internet of Things (IoT) devices and scenarios, e.g. for the ‘smart’ home, and with different groups of co-
+designers [3,7,8]. The Loaded Dice filled a gap between analog, non-functional tools, often card-based, e.g. KnowCards 
+[1], and functional but tinkering based tools, e.g. littleBits [2], for multisensory und multimodal exploration, ideation 
+and prototyping.
+We introduce the Loaded Dice, the core concepts that they are built on, the used sensors and actuators, and how 
+they map to different human senses. We will then continue to discuss how we realized mappings between sensed raw 
+value, normalized intermediate values, and actuated values. While the mappings that we currently use are sufficiently 
+good enough for current purposes, we see big potential in some extended uses as ‘Synesthetic Dice’.
+This brings us to our core question:  How can the Loaded Dice be used for exploration and research of synesthetic 
+mappings between sensors and actuators, e.g. for innovative interactions and non-verbal communication?
+
+SENSORDIE
+TemperatureSensor
+Light Sensor
+Microphone
+MovementSensor
+Potentiometer
+Distance Sensor
+ACTUATORDIE
+Vibration
+Heating Surface
+LED-Bargraph
+Loudspeaker
+Power-LEDs
+FanFigure 1: The Loaded Dice; left: example of devices in use, turning heat into light (sensor die with temperature sensor active and 
+actuator die with power LED active) [9]; right faces and functions – sensors and actuators [8]
+2
+THE LOADED DICE - SENSES, SENSORS, ACTUATORS
+The Loaded Dice1 are a set of two cubical devices wirelessly connected (fig. 2a). Each cube has six sides, offering in one 
+cube six sensors and in the other cube six actuators, one on each side, suitable for multisensory and multimodal 
+environmental and user interactions. The sensor cube normalizes a raw sensor value meaningfully, transmits it, and 
+then the other cube actuates it mapped on an output. The cubical shape communicates the intuitive reading that the 
+top side is active, like a die, offering an easy and spontaneous way to re-combine sensors and actuators. Every sensor-
+in and actuator-out combination is possible resulting in 36 combinations in total. [5] 
+The “traditional five” human senses are sight, hearing, taste, smell and touch. Secondary senses are temperature, 
+pain, proprioception and balance. Due to the constraints of the technical platform we could not address all human 
+senses with sensors and actuators. Overall the Loaded Dice holds sensors and actuators equivalent to some human 
+senses directly (see fig. 1 and table 1 for details). It is also possible to think about effects to address other senses using  
+the given sensors and actuators, e.g. to inflict pain via the Peltier element through excessive heat or cold (not intended 
+nor recommended). It is also possible (but currently not implemented) to use the internal inertial measurement unit 
+(IMU), consisting of an accelerometer and gyrometer, not only for interaction controls but also as a sense, as an  
+equivalent to proprioception and balance (movement and position). 
+New multisensory interaction modalities are possible but not yet implemented, e.g. olfactory / smell. They have the 
+potential to broaden interaction qualities even further and especially in an emotional way [6].
+Table 1. Human senses vs. sensors and actuators in the Loaded Dice
+Human Sense
+Sensor
+Actuator
+sight
+(visual stimuli)
+luxmeter (visible light luminosity/ brightness)
+passive infrared detector (PIR movement)
+ultrasonic transceiver (distance)
+power LED (brightness)
+LED ring-graph (count, overall brightness, color)
+hearing
+(auditive stimuli)
+microphone (amplitude)
+sound (modulated note for instrument)
+(vibration motor, rattling noise)
+(fan, air flow noise)
+touch
+(tactile stimuli)
+potentiometer (manual angular dial of 270°)
+vibration motor (vibration)
+fan (mechanical stimulation on hairs)
+temperature
+(thermal stimuli)
+infrared thermometer (thermopile / thermal 
+radiation)
+Peltier element (cooling and heating plate)
+fan (cooling by chill effect on skin)
+1 video demonstrating the Loaded Dice: https://www.youtube.com/watch?v=-E5aUiktCic
+2
+
+SENSORDIE
+TemperatureSensor
+Light Sensor
+Microphone
+MovementSensor
+Potentiometer
+Distance Sensor
+ACTUATORDIE
+Vibration
+Heating Surface
+LED-Bargraph
+Loudspeaker
+Power-LEDs
+Fan3
+SYNESTHESIA - MAPPING SENSES AND MODALITIES
+Synesthesia describes the phenomenon of an event being experienced by another, separate sensory modality [4]. While 
+medical not exact, in principle, this means a sound might not only be heard but also be seen as a color (as an example). 
+Most existing tools, i.e. for IoT ideation, do not employ synesthesia effects as a design opportunity in order to break  
+with existing sensing stereotypes for framing design spaces. Such a stereotype could be e.g. that making noise should 
+always  be  connected  with  hearing  noise.  While  most  related  digital  (IoT)  ideation  tools  do allow  for flexible 
+combinations of different sensors and actuators in principle, this is not ad hoc possible. Instead they require necessary 
+steps in combining parts or mapping sensor values to actuator values. Thus, they demand an initial idea of how the 
+combination should play out. Our tool allows users to explore such synesthetic effects ad hoc.
+We implemented a meaningful mapping between every sensor and actuator that is used in the Loaded Dice. This 
+includes reasonably chosen sampling rates, ranges and steppings for raw input values, their normalization on internal 
+values and the conversion back to meaningful output values. All this is done internally in hard- and software, without 
+the need of user intervention. Selecting a new sensor-actuator combination just requires bringing another side to the 
+top. Based on the presented design rationale, a co-designer can transport heat over a distance by choosing the infrared 
+thermometer and Peltier element sides of both cubes. Rotating the actuator cube to the power-LEDs would transform 
+the temperature into light, thus mimicking synesthesia-like perception.
+The possibilities  of the  Loaded Dice can be used in a framed scenario-driven co-design approach, in open 
+exploration or even just for ‘sensory sketching’, even for ‘weird’ synesthetic combinations, e.g:
+
+to try out how bright sunlight sounds or feels as vibration
+
+what temperature a loud cry has
+
+how much air-flow half a meter distance is
+
+whether you can feel the flickering of light …
+We  use  meaningful  but  simple  functions  for  preprocessing  of  raw  sensor  values  and  normalization  to  an 
+intermediate data value as well back to actuations (table 2). Overall, the mappings are done in a predefined ‘static’  
+way. However, static does not mean one fits all. It is necessary to consider non-linearities and dynamics, e.g. for light  
+and sound, as these senses are not perceived in a linear or static manner by humans. However, we applied ‘just good  
+enough’ assumptions for meaningfulness without the claim of physical or psychometric correctness, sometimes even a 
+bit off to make effects clearer. Currently also the sensor as well as the selected actuator are considered for the mapping 
+in addition to the normalized value. We do this mainly for technical reasons as the different modalities operate at 
+different speeds. Currently, only the LED ring graphic signals which sensor has sampled the data by changing color.
+Table 2: Current mapping from sensed values to intermediate values and then to actuated values
+Sensor
+Sensor Mapping 
+Value
+Actuator Mapping
+Actuator
+potentiometer
+0..270° AD sampling 0..1023  linear  0..24
+0..24
+ Neopixels count 0..24, color coded by sensor, 
+brightness per pixel static
+ring-graph
+thermometer
+digital read-out 0..50 °C  linear  0..24
+ sqr  0..576  0..255 RGB brightness
+power LED
+microphone
+50ms window AD sampling 0..1023  max-min 
+difference  0..1023  linear  0..24
+0  0; 1..24  MIDI noteOn(value+50) 
+sound
+distance
+0  0; 1..72 cm  linear  1..24
+0..12  -255..0 (cooling) 12..24  0..255 (heating) PWM
+or 0..24  0..255 (from neutral to heating only) PWM
+Peltier thermo
+PIR movement
+binary 0  0; 1  24
+0  0; 1..24  64..255 PWM
+vibration
+light
+digital read-out 0..65535 lx  sqrt  0..48  0..24
+0  0; 1..24  160..255 PWM
+fan
+Every combination is possible, alignment in lines just as examples. AD: analogdigital conversion, PWM: pulse width modulation
+3
+
+SENSORDIE
+TemperatureSensor
+Light Sensor
+Microphone
+MovementSensor
+Potentiometer
+Distance Sensor
+ACTUATORDIE
+Vibration
+Heating Surface
+LED-Bargraph
+Loudspeaker
+Power-LEDs
+FanWhile we are quite satisfied what the Loaded Dice can already do there are some new possibilities at hand:
+
+more use of colors: for power LED element and LED ring-graph (NeoPixels are colorful…)
+
+other use of sound: other (music/midi) instruments, modulation of velocity and pitch, other sounds 
+(artificial or sampled in nature)
+
+use of spatial component: position of the LEDs of the ring-graph, color fades, patterns
+
+use of temporal components: from time static value to dynamic patterns for sound, vibration, light, air 
+flow etc.
+A flexible “sketching” of a new mapping function would allow to bring in completely new synesthesia effects, also not 
+necessarily only limited to one input sensor and one output actuator at one time.
+4
+CONCLUSION
+While the Loaded Dice can already be used meaningfully for activities associated with synesthesia, e.g. for ideation, we 
+see a lot of potential in more flexible mappings and even other creative uses of what the sensors and actuators might 
+do. We are open for inspirations and ideas.
+ACKNOWLEDGMENTS
+This research is funded by the German Ministry of Education and Research (BMBF), grant FKZ 16SV7116.
+References
+[1]
+Tina Aspiala and Alexandra Deschamps-Sonsino. 2016. Know Cards: Learn. Play. Collect.  Know Cards. Retrieved December 6, 2016 from 
+http://know-cards.myshopify.com/
+[2]
+Ayah Bdeir. 2009. Electronics As Material: LittleBits. In Proceedings of the 3rd International Conference on Tangible and Embedded Interaction (TEI 
+’09), 397–400. https://doi.org/10.1145/1517664.1517743
+[3]
+Arne Berger, William Odom, Michael Storz, Andreas Bischof, Albrecht Kurze, and Eva Hornecker. 2019. The Inflatable Cat: Idiosyncratic Ideation 
+Of Smart Objects For The Home. In CHI Conference on Human Factors in Computing Systems Proceedings. https://doi.org/10.1145/3290605.3300631
+[4]
+Peter G. Grossenbacher and Christopher T. Lovelace. 2001. Mechanisms of synesthesia: cognitive and physiological constraints. Trends in cognitive 
+sciences 5, 1: 36–41. Retrieved December 15, 2016 from http://www.sciencedirect.com/science/article/pii/S1364661300015710
+[5]
+Albrecht Kurze. 2021. Interaction Qualities For Interactions With, Between, And Through IoT Devices. In 11th International Conference on the 
+Internet of Things (IoT ‘21), November 08-12, 2021, St.Gallen, Switzerland. https://doi.org/10.1145/3494322.3494348
+[6]
+Albrecht Kurze. 2021. Scented Dice: New interaction qualities for ideating connected devices. In Workshop Smell, Taste, and Temperature Interfaces 
+at Conference on Human Factors in Computing Systems (CHI ’21). Retrieved from https://arxiv.org/abs/2201.10484
+[7]
+Albrecht Kurze, Kevin Lefeuvre, Michael Storz, Andreas Bischof, Sören Totzauer, and Arne Berger. 2016. Explorative Co-Design-Werkzeuge zum 
+Entwerfen von Smart Connected Things am Beispiel eines Workshops mit Blinden und Sehbehinderten. In Technische Unterstützungssysteme, die 
+die Menschen wirklich wollen, 395–400. Retrieved January 19, 2017 from http://tinyurl.com/janya26
+[8]
+Kevin Lefeuvre, Sören Totzauer, Andreas Bischof, Albrecht Kurze, Michael Storz, Lisa Ullmann, and Arne Berger. 2016. Loaded Dice: Exploring the 
+Design Space of Connected Devices with Blind and Visually Impaired People. In Proceedings of the 9th Nordic Conference on Human-Computer 
+Interaction (NordiCHI ’16), 31:1-31:10. https://doi.org/10.1145/2971485.2971524
+[9]
+Kevin Lefeuvre, Sören Totzauer, Andreas Bischof, Michael Storz, Albrecht Kurze, and Arne Berger. 2017. Loaded Dice: How to cheat your way to 
+creativity. In Proceedings of the 3rd Biennial Research Through Design Conference. https://doi.org/10.6084/m9.figshare.4746976.v1
+4
+
+SENSORDIE
+TemperatureSensor
+Light Sensor
+Microphone
+MovementSensor
+Potentiometer
+Distance Sensor
+ACTUATORDIE
+Vibration
+Heating Surface
+LED-Bargraph
+Loudspeaker
+Power-LEDs
+Fan

GdFJT4oBgHgl3EQfECxK/content/tmp_files/load_file.txt

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+page_content='Synesthetic Dice: Sensors, Actuators, And Mappings  Albrecht Kurze  Chemnitz University of Technology, Albrecht.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content='Kurze@informatik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content='tu-chemnitz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content='de  How bright can you cry?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content=' How loud does the sun shine?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content=' We developed a multisensory and multimodal tool, the Loaded Dice, for use in  co-design workshops to research the design space of IoT usage scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content=' The Loaded Dice incorporate the principle of a technical  synesthesia, being able to map any of the included sensors to any of the included actuators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content=' With just a turn of one of the cubical  devices it is possible to create a new combination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content=' We discuss the core principles of the Loaded Dice, what sensors and actuators are  included, how they relate to human senses, and how we realized a meaningful mapping between sensors and actuators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content=' We further  discuss where we see additional potential in the Loaded Dice to support synesthetic exploration – as Synesthetic Dice – so that you can  eventually find out who cries brighter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content='  CCS CONCEPTS • Human-centered computing~Human computer interaction (HCI)  Additional  Keywords  and  Phrases:  multisensory,  multimodal,  synesthesia,  design,  ideation,  tools,  methods,  IoT,  Internet of Things, haptic technology, cubic shape, tangible interactive devices, input and output devices, tangibles  ACM Reference Format:  Albrecht Kurze.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content=' Synesthetic Dice: Sensors, Actuators, And Mappings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content=' In Workshop Sensory Sketching (CHI’22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content='  April 22, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content=' 4 pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content='  1  INTRODUCTION  Some years ago we designed and developed the Loaded Dice [8,9], a multisensory and multimodal hybrid toolkit to  ideate Internet of Things (IoT) devices and scenarios, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content=' for the ‘smart’ home, and with different groups of co-  designers [3,7,8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content=' The Loaded Dice filled a gap between analog, non-functional tools, often card-based, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content=' KnowCards  [1], and functional but tinkering based tools, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content=' littleBits [2], for multisensory und multimodal exploration, ideation  and prototyping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content='  We introduce the Loaded Dice, the core concepts that they are built on, the used sensors and actuators, and how  they map to different human senses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content=' We will then continue to discuss how we realized mappings between sensed raw  value, normalized intermediate values, and actuated values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content=' While the mappings that we currently use are sufficiently  good enough for current purposes, we see big potential in some extended uses as ‘Synesthetic Dice’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content='  This brings us to our core question:  How can the Loaded Dice be used for exploration and research of synesthetic  mappings between sensors and actuators, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content=' for innovative interactions and non-verbal communication?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content='  SENSORDIE  TemperatureSensor  Light Sensor  Microphone  MovementSensor  Potentiometer  Distance Sensor  ACTUATORDIE  Vibration  Heating Surface  LED-Bargraph  Loudspeaker  Power-LEDs  FanFigure 1: The Loaded Dice;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content=' left: example of devices in use, turning heat into light (sensor die with temperature sensor active and  actuator die with power LED active) [9];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content=' right faces and functions – sensors and actuators [8]  2  THE LOADED DICE - SENSES, SENSORS, ACTUATORS  The Loaded Dice1 are a set of two cubical devices wirelessly connected (fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content=' 2a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content=' Each cube has six sides, offering in one  cube six sensors and in the other cube six actuators, one on each side, suitable for multisensory and multimodal  environmental and user interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content=' The sensor cube normalizes a raw sensor value meaningfully, transmits it, and  then the other cube actuates it mapped on an output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content=' The cubical shape communicates the intuitive reading that the  top side is active, like a die, offering an easy and spontaneous way to re-combine sensors and actuators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content=' Every sensor-  in and actuator-out combination is possible resulting in 36 combinations in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content=' [5]  The “traditional five” human senses are sight, hearing, taste, smell and touch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content=' Secondary senses are temperature,  pain, proprioception and balance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content=' Due to the constraints of the technical platform we could not address all human  senses with sensors and actuators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content=' Overall the Loaded Dice holds sensors and actuators equivalent to some human  senses directly (see fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content=' 1 and table 1 for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content=' It is also possible to think about effects to address other senses using  the given sensors and actuators, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content=' to inflict pain via the Peltier element through excessive heat or cold (not intended  nor recommended).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content=' It is also possible (but currently not implemented) to use the internal inertial measurement unit  (IMU), consisting of an accelerometer and gyrometer, not only for interaction controls but also as a sense, as an  equivalent to proprioception and balance (movement and position).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content='  New multisensory interaction modalities are possible but not yet implemented, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content=' olfactory / smell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content=' They have the  potential to broaden interaction qualities even further and especially in an emotional way [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content='  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content=' Human senses vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content=' sensors and actuators in the Loaded Dice  Human Sense  Sensor  Actuator  sight  (visual stimuli)  luxmeter (visible light luminosity/ brightness)  passive infrared detector (PIR movement)  ultrasonic transceiver (distance)  power LED (brightness)  LED ring-graph (count,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content=' overall brightness,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content=' color)  hearing  (auditive stimuli)  microphone (amplitude)  sound (modulated note for instrument)  (vibration motor,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content=' rattling noise)  (fan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content=' air flow noise)  touch  (tactile stimuli)  potentiometer (manual angular dial of 270°)  vibration motor (vibration)  fan (mechanical stimulation on hairs)  temperature  (thermal stimuli)  infrared thermometer (thermopile / thermal  radiation)  Peltier element (cooling and heating plate)  fan (cooling by chill effect on skin)  1 video demonstrating the Loaded Dice: https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content='youtube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content='com/watch?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content='v=-E5aUiktCic  2  SENSORDIE  TemperatureSensor  Light Sensor  Microphone  MovementSensor  Potentiometer  Distance Sensor  ACTUATORDIE  Vibration  Heating Surface  LED-Bargraph  Loudspeaker  Power-LEDs  Fan3  SYNESTHESIA - MAPPING SENSES AND MODALITIES  Synesthesia describes the phenomenon of an event being experienced by another, separate sensory modality [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content=' While  medical not exact, in principle, this means a sound might not only be heard but also be seen as a color (as an example).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content='  Most existing tools, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content=' for IoT ideation, do not employ synesthesia effects as a design opportunity in order to break  with existing sensing stereotypes for framing design spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content=' Such a stereotype could be e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content=' that making noise should  always  be  connected  with  hearing  noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content=' While  most  related  digital  (IoT)  ideation  tools  do allow  for flexible  combinations of different sensors and actuators in principle, this is not ad hoc possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content=' Instead they require necessary  steps in combining parts or mapping sensor values to actuator values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content=' Thus, they demand an initial idea of how the  combination should play out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content=' Our tool allows users to explore such synesthetic effects ad hoc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content='  We implemented a meaningful mapping between every sensor and actuator that is used in the Loaded Dice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content=' This  includes reasonably chosen sampling rates, ranges and steppings for raw input values, their normalization on internal  values and the conversion back to meaningful output values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content=' All this is done internally in hard- and software, without  the need of user intervention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content=' Selecting a new sensor-actuator combination just requires bringing another side to the  top.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content=' Based on the presented design rationale, a co-designer can transport heat over a distance by choosing the infrared  thermometer and Peltier element sides of both cubes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content=' Rotating the actuator cube to the power-LEDs would transform  the temperature into light, thus mimicking synesthesia-like perception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content='  The possibilities  of the  Loaded Dice can be used in a framed scenario-driven co-design approach, in open  exploration or even just for ‘sensory sketching’, even for ‘weird’ synesthetic combinations, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content='g: to try out how bright sunlight sounds or feels as vibration what temperature a loud cry has how much air-flow half a meter distance is whether you can feel the flickering of light …  We  use  meaningful  but  simple  functions  for  preprocessing  of  raw  sensor  values  and  normalization  to  an  intermediate data value as well back to actuations (table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content=' Overall, the mappings are done in a predefined ‘static’  way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content=' However, static does not mean one fits all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content=' It is necessary to consider non-linearities and dynamics, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content=' for light  and sound, as these senses are not perceived in a linear or static manner by humans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content=' However, we applied ‘just good  enough’ assumptions for meaningfulness without the claim of physical or psychometric correctness, sometimes even a  bit off to make effects clearer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content=' Currently also the sensor as well as the selected actuator are considered for the mapping  in addition to the normalized value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content=' We do this mainly for technical reasons as the different modalities operate at  different speeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content=' Currently, only the LED ring graphic signals which sensor has sampled the data by changing color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content='  Table 2: Current mapping from sensed values to intermediate values and then to actuated values  Sensor  Sensor Mapping  Value  Actuator Mapping  Actuator  potentiometer  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
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+page_content='.72 cm \uf0e0 linear \uf0e0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content='.24  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
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+page_content='.0 (cooling) 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content='.24 \uf0e0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content='.255 (heating) PWM  or 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content='.24 \uf0e0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
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+page_content='.24 \uf0e0 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content='.255 PWM  vibration  light  digital read-out 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content='.65535 lx \uf0e0 sqrt \uf0e0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content='.48 \uf0e0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content='.24  0 \uf0e0 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content='.24 \uf0e0 160.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content='.255 PWM  fan  Every combination is possible, alignment in lines just as examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content=' AD: analog\uf0e0digital conversion,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content=' PWM: pulse width modulation  3  SENSORDIE  TemperatureSensor  Light Sensor  Microphone  MovementSensor  Potentiometer  Distance Sensor  ACTUATORDIE  Vibration  Heating Surface  LED-Bargraph  Loudspeaker  Power-LEDs  FanWhile we are quite satisfied what the Loaded Dice can already do there are some new possibilities at hand: more use of colors: for power LED element and LED ring-graph (NeoPixels are colorful…) other use of sound: other (music/midi) instruments,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content=' modulation of velocity and pitch,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content=' other sounds  (artificial or sampled in nature) use of spatial component: position of the LEDs of the ring-graph,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content=' color fades,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content=' patterns use of temporal components: from time static value to dynamic patterns for sound,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content=' vibration,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content=' light,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content=' air  flow etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content='  A flexible “sketching” of a new mapping function would allow to bring in completely new synesthesia effects, also not  necessarily only limited to one input sensor and one output actuator at one time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content='  4  CONCLUSION  While the Loaded Dice can already be used meaningfully for activities associated with synesthesia, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content=' for ideation, we  see a lot of potential in more flexible mappings and even other creative uses of what the sensors and actuators might  do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content=' We are open for inspirations and ideas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content='  ACKNOWLEDGMENTS  This research is funded by the German Ministry of Education and Research (BMBF), grant FKZ 16SV7116.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content='  References  [1]  Tina Aspiala and Alexandra Deschamps-Sonsino.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content=' Know Cards: Learn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content=' Play.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content=' Collect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content=' Know Cards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content=' Retrieved December 6, 2016 from  http://know-cards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content='myshopify.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content='com/  [2]  Ayah Bdeir.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
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+page_content=' Electronics As Material: LittleBits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content=' In Proceedings of the 3rd International Conference on Tangible and Embedded Interaction (TEI  ’09), 397–400.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content='1145/1517664.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
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+page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content=' The Inflatable Cat: Idiosyncratic Ideation  Of Smart Objects For The Home.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content=' In CHI Conference on Human Factors in Computing Systems Proceedings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
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+page_content=' Grossenbacher and Christopher T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content=' Lovelace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
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+page_content=' Trends in cognitive  sciences 5, 1: 36–41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content=' Retrieved December 15, 2016 from http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content='sciencedirect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content='com/science/article/pii/S1364661300015710  [5]  Albrecht Kurze.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content=' Interaction Qualities For Interactions With, Between, And Through IoT Devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content=' In 11th International Conference on the  Internet of Things (IoT ‘21), November 08-12, 2021, St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content='Gallen, Switzerland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
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+page_content=' In Workshop Smell, Taste, and Temperature Interfaces  at Conference on Human Factors in Computing Systems (CHI ’21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content=' Retrieved from https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
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+page_content='10484  [7]  Albrecht Kurze, Kevin Lefeuvre, Michael Storz, Andreas Bischof, Sören Totzauer, and Arne Berger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content=' Explorative Co-Design-Werkzeuge zum  Entwerfen von Smart Connected Things am Beispiel eines Workshops mit Blinden und Sehbehinderten.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content=' In Technische Unterstützungssysteme, die  die Menschen wirklich wollen, 395–400.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content=' Retrieved January 19, 2017 from http://tinyurl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content='com/janya26  [8]  Kevin Lefeuvre, Sören Totzauer, Andreas Bischof, Albrecht Kurze, Michael Storz, Lisa Ullmann, and Arne Berger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content=' Loaded Dice: Exploring the  Design Space of Connected Devices with Blind and Visually Impaired People.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content=' In Proceedings of the 9th Nordic Conference on Human-Computer  Interaction (NordiCHI ’16), 31:1-31:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content='1145/2971485.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content='2971524  [9]  Kevin Lefeuvre, Sören Totzauer, Andreas Bischof, Michael Storz, Albrecht Kurze, and Arne Berger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content=' Loaded Dice: How to cheat your way to  creativity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content=' In Proceedings of the 3rd Biennial Research Through Design Conference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content='6084/m9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content='figshare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content='4746976.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}
+page_content='v1  4  SENSORDIE  TemperatureSensor  Light Sensor  Microphone  MovementSensor  Potentiometer  Distance Sensor  ACTUATORDIE  Vibration  Heating Surface  LED-Bargraph  Loudspeaker  Power-LEDs  Fan' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdFJT4oBgHgl3EQfECxK/content/2301.11436v1.pdf'}

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@@ -0,0 +1,1507 @@
+Age-Optimal Multi-Channel-Scheduling under
+Energy and Tolerance Constraints
+Xujin Zhou, Irem Koprulu, Atilla Eryilmaz
+Electrical and Computer Engineering
+The Ohio State University
+Columbus, US
+{zhou.2400@osu.edu, irem.koprulu@gmail.com, eryilmaz.2@osu.edu}
+Abstract—We study the optimal scheduling problem where n
+source nodes attempt to transmit updates over L shared wireless
+on/off fading channels to optimize their age performance under
+energy and age-violation tolerance constraints. Specifically, we
+provide a generic formulation of age-optimization in the form of
+a constrained Markov Decision Processes (CMDP), and obtain
+the optimal scheduler as the solution of an associated Linear
+Programming problem. We investigate the characteristics of the
+optimal single-user multi-channel scheduler for the important
+special cases of average-age and violation-rate minimization.
+This leads to several key insights on the nature of the optimal
+allocation of the limited energy, where a usual threshold-based
+policy does not apply and will be useful in guiding scheduler
+designers. We then investigate the stability region of the optimal
+scheduler for the multi-user case. We also develop an online
+scheduler using Lyapunov-drift-minimization methods that do
+not require the knowledge of channel statistics. Our numerical
+studies compare the stability region of our online scheduler to the
+optimal scheduler to reveal that it performs closely with unknown
+channel statistics.
+I. INTRODUCTION
+In recent years, the Internet of Things (IoT) has become
+one of the most important frameworks of the next-generation
+wireless networks, whereby a large number of mobile devices
+need to be supported over an ultra-wide frequency spectrum
+(see, for example, [1]). In particular, for many real-time IoT
+applications, it is necessary for the devices to send fresh
+updates over the shared spectrum. To measure the freshness
+of data, the concept of Age of Information (AoI) has been
+introduced over the last decade (see, for example, [2]–[4]),
+which is defined concisely as the elapsed time since the
+generation time of the last received status update. Since
+the introduction of the AoI metric, numerous related studies
+emerged in various networking scenarios, including wireless
+random access networks (e.g., [5], [6]), content distribution
+networks (e.g., [7], [8]), scheduling (e.g., [9]–[13]), queuing
+networks (e.g., [14], [15]), and vehicular networks (e.g., [16]).
+Recently, other AoI related metrics have been developed in
+order to address more generalized or different forms of ageing,
+such as: non-linear AoI (e.g., [4], [17]), peak AoI (e.g., [18]),
+time-since-last-service (e.g., [19]), age upon decisions (e.g.,
+[20]), to name a few. Among them, the metric, called the age-
+violation-rate (see [15], [21], [22]) is of particular interest for
+real-time IoT services that have hard age-deadline constraints
+and a limited tolerance to violating this deadline (see [23],
+[24] for further motivation of this metric).
+In view of its significance for next generation IoT networks,
+in this paper, we study the general optimal multi-channel
+scheduling problem to optimize varying forms of age perfor-
+mances under energy and age-violation tolerance constraints.
+Our contributions can be listed as:
+• We provide a generic formulation of age-optimization
+as a Constrained Markov Decision Problem (CMDP)
+(see [25]–[27]) and obtain the age-optimal multi-channel
+scheduler as the solution of an associated Linear Pro-
+gramming problem, first for the single-source (in Sec-
+tion III) and then for general the multi-source (in Sec-
+tion IV) scenarios.
+• For the single-source multi-channel scenario, we in-
+vestigate the characteristics of the optimal schedulers
+under energy constraints for two age metrics that are
+important for IoT applications: (i) average-age mini-
+mization; and (ii) age-violation-rate minimization, a non-
+convex/concave metric (in Section III-C). Our investiga-
+tions reveal various insights on different energy allocation
+structures, as well as the common monotonicity proper-
+ties of the optimal schedulers for minimizing these two
+metrics, which is useful for guiding scheduler designers.
+• For the multi-source age optimal scheduling problem,
+we also study the feasibility region of the average-
+age-optimal scheduler under age-violation-rate tolerance
+constraints to contrast its results with those of related
+earlier works that are developed for the single-channel
+multi-user scenario (see Section IV-C and Section VI).
+• Moreover, we develop (in Section V) an online scheduler
+using Lyapunov-drift-minimization methods (e.g., [28])
+that does not require the knowledge of channel statistics,
+and compare its performance to the optimal and earlier
+designs to reveal how much the knowledge of channel
+statistics affects the feasibility region (see Section VI).
+Our work relates to, but also differs from several other
+related works in this domain. Many early works (e.g., [9],
+[12], [29]) aim to minimize AoI under power constraints
+but with the assumption of reliable channels as opposed to
+the fading channels that we consider. More recent works
+arXiv:2301.00562v1  [cs.IT]  2 Jan 2023
+
+(e.g., [10], [30]) aim to minimize AoI-related costs based
+on max-age matching, while other works (e.g., [29], [31])
+proposed AoI minimization schedulers based on Whittle Index
+approach. However, to the best of our knowledge, prior works
+predominantly assume that one source can choose at most
+one channel, which is an important factor in proving the
+Whittle Indexability of the corresponding problems they solve.
+In contrast, one of the key features our setting is the possibility
+of each user to transmit over multiple channels as enabled by
+new wireless technologies. Furthermore, most of the above
+mentioned works have average or peak AoI as the objective
+function, while we consider more general age-based objective
+functions, which for example allows the objective function to
+be a non-convex metric such as the age-violation-rate. In this
+multi-channel setting with general objectives, we observe (cf.
+Section III-C) that the optimal solution can in fact possess non-
+monotone characteristics, which make the Whittle Indexability
+approach infeasible in general. The work in [21] has con-
+sidered the multi-source single-channel scheduling problem
+under tolerance constraints, which is a special case of our
+setting. We would like to note that this interesting work
+[21] has been a primary motivation for our current work in
+exploring a different approach based on the CMDP framework
+that guarantees optimality and applies to more general multi-
+channel scenarios with additional energy constraints. There
+are also works (e.g., [32], [33]) that focus on learning-based
+approaches which can be considered as complementary to the
+focus of this work.
+II. SYSTEM MODEL
+We consider the operation of a discrete-time wireless access
+system, whereby n source nodes share L on/off fading wireless
+channels to update their ageing status at a receiver (such as a
+base station) under energy and violation tolerance constraints
+(see Figure 1).
+Figure 1. n sources share L on-off fading channels to update their status to
+a receiver under energy and tolerance constraints in order to keep their age
+levels low.
+Our goal is to develop generic solution strategies to find
+optimal schedulers that can optimize diverse age-based metrics
+while meeting certain requirements on energy consumption
+and tolerance levels. We describe the key terminology and the
+essential system dynamics in the rest of this section. Then,
+in the following sections we formulate and solve classes of
+age-optimization problems for single and multi-source cases,
+subsequently.
+Scheduling policy and age-violation-tolerance: We assume
+that each source node i ∈ {1, · · · , n} refreshes its status
+and creates a new packet at the beginning of every time
+slot t ∈ {1, 2, 3, · · · }. Source nodes attempt to transmit their
+freshest packet to the receiver, for example a base station
+(BS), whenever they get a chance to transmit. Every time the
+BS successfully receives a new status from source node i,
+it saves the current status and discards all previous packets
+received from that node. As such, the BS keeps only one
+packet from each source node, namely the freshest one. We
+use Xi[t] to denote the generation time of the packet stored
+at the BS from source i at time t. We define the age Ai[t]
+of source node i at time t as the time that has elapsed since
+the generation of its last received packet1: Ai[t] ≜ t − Xi[t].
+We use2 A[t] ≜ (A1[t], · · · , An[t]) to denote the ages of all
+sources at time slot t.
+At the beginning of each time slot, the centralized scheduler
+decides which channels each of the source nodes will use to
+transmit to the base station based on the ages A[t] of all source
+nodes. Let ui(A[t]) be the number of channels source node
+i uses to transmit at time t. Each transmission attempt can
+resolve in success or failure which we will describe below
+as part of the channel success model. If the base station
+successfully receives the packet from source i at time t, then
+its age at time t + 1 will reset to 1, otherwise its age will
+increase by one, i.e.,
+Ai[t + 1] =
+�
+1, if transmission of source i succeeds
+Ai[t] + 1,
+otherwise.
+We allow each source i to have a desired age thresh-
+old/deadline τi. The information of source i is up-to-date if
+its age is less than or equal to this threshold τi. Otherwise, we
+speak of an age violation in that slot. In particular, we define
+the age-violation-rate of source i as the long-term average
+fraction of time slots when the source’s age Ai[t] exceeds
+its threshold τi, i.e., lim
+T →∞
+1
+T
+T
+�
+t=1
+1 {Ai[t] > τi}. We use ϵi ∈
+[0, 1] to indicate the tolerance of source i that measures the
+maximum allowed age-violation-rate for its updates. (ϵi = 1
+indicates that there is no violation rate constraint, and ϵi = 0
+indicates that we do not allow any deadline violation.) When
+the age violation rate is no greater than the tolerance rate, the
+age violation tolerance constraint is satisfied.
+Channel success model and energy constraints: The n
+source nodes share L wireless on/off fading channels, each
+of which can accommodate at most one packet transmission.
+However, even when there is a single transmission over
+a channel, a successful transmission is not guaranteed. In
+1This metric is also referred to as Age-of-Information (AoI) and Time-Since-
+Last-Service (TSLS) in different contexts. In the rest of the paper, we will refer
+to it as AoI or simple as age, interchangeably.
+2We will consistently use bold symbols to represent vectors.
+
+particular, source node i has a channel success probability of
+µi when transmitting over each of its assigned channels3.
+We call the update of source i in a slot to be a success
+if any one of its transmissions over its assigned channels is
+successful. Since the channel is a collision channel, for an
+optimal scheduler we always have
+n
+�
+i=1
+ui(A[t]) ≤ L. Once
+the value of ui(A[t]) is decided for all i, the scheduler will
+assign different channels to different sources, so that no two
+sources transmit over the same channel. Also, note that under
+the described channel success model, the probability for the
+BS to successfully receive an update from source node i when
+the node uses l channels is 1 − (1 − µi)l.
+We assume that each transmission over a channel comes
+with an energy cost of 1 unit4. We require that the aggregate
+time-average energy cost for source i is not greater than a
+given constraint bi channels per slot, i.e., we require
+lim
+T →∞
+1
+T
+T
+�
+t=1
+ui (A[t]) ≤ bi,
+bi ∈ R+.
+It is obvious that transmitting over more channels will
+increase the success probability of a source, but increase
+energy consumption. We are interested in finding the number
+of channels that when allocated to sources optimize the desired
+age performance given the current age state, as well as energy
+and and tolerance constraints discussed above. In the next
+section, we attack this problem within the constrained Markov
+Decision Process (MDP) framework first for a single user, and
+then extend our approach to cover the multi-user setting.
+III. AGE-OPTIMAL MULTI-CHANNEL SCHEDULING FOR A
+SINGLE USER
+In this section, we first consider the single-user age-optimal
+multi-channel scheduling problem. This not only allows us to
+simplify the notation by omitting the subscripts, but also is of
+particular interest for the next generation ultra-wideband wire-
+less communication technologies that are expected to support
+low-delay access over multiple fading channels. We formulate
+a general age-optimal optimization problem which can be
+used in different scenarios in Section III-A and following the
+analysis of the performance in Section III-B. To that end,
+in Section III-C, we study the characterization and insights
+of the optimal schedulers for two important special cases of
+minimizing the average-age and the age-violation-rate, which
+will be useful in guiding scheduler designers.
+A. Problem formulation
+The problem of minimizing time-averaged age-based ob-
+jectives under average energy and tolerance constraints can
+3All our development can be generalized to the case when the success
+probability between source i and channel j is allowed to be different as µij.
+However, this is omitted here as it increases the complexity of the exposition
+without adding to the substance.
+4This can also be generalized to non-uniform energy costs over different
+channels, but omitted to avoid cumbersome notation.
+be generally formulated as the following constrained Markov
+decision problem [25]:
+min
+u(A)
+lim
+T →∞
+1
+T
+T
+�
+t=1
+E [ω0(A[t])]
+(1)
+s.t :
+lim
+T →∞
+1
+T
+T
+�
+t=1
+E [u (A[t])] ≤ b,
+(2)
+lim
+T →∞
+1
+T
+T
+�
+t=1
+E [ωk (A[t])] ≤ ck, k = 1, · · · , K,
+u(A[t]) ∈ {0, 1, · · · , L}.
+The optimization is performed over Markovian policies
+described by a function u(·) that maps age levels to number
+of channels. It is known that such Markovian policies are
+sufficient for optimal operation [25].
+The first constraint on the time-averaged u(·) captures the
+average energy constraint discussed in the system model. The
+functions ωk(·) serve as general functions that map the current
+state A[t] to a value that measures the cost of that age with
+respect to various measures5 By setting different mappings for
+the weight function ω0(A[t]), the objective can be changed
+into different commonly used age-related objectives: letting
+ω0(a) = −1{a = 1} transform the objective to maximizing
+the average throughput; letting ω0(a) = a makes the objective
+minimize the average AoI; letting ω0(a) = 1{a ≥ d} make the
+objective minimize the average age-violation rate. Note that
+this allows the objective function to be a non-convex/concave
+function.
+B. Performance analysis
+Next, we will analyze the generic constrained optimization
+problem under energy constraint by showing that the problem
+is equivalent to a Linear Programming (LP) problem and thus
+describe the optimal policy.
+Theorem 1: The solution of the generic age-optimization
+problem (1) can be obtained by solving the following linear
+programming problem:
+min
+yla
+D
+�
+a=1
+L
+�
+l=0
+yl
+aω0(a)
+s.t:
+D
+�
+a=1
+L
+�
+l=0
+yl
+a · l ≤ b,
+D
+�
+a=1
+L
+�
+l=0
+yl
+aωk(a) ≤ ck, k = 1, · · · , K,
+0 ≤ yl
+a ≤ 1
+∀1 ≤ a ≤ D, 0 ≤ l ≤ L,
+D
+�
+a=1
+L
+�
+l=0
+yl
+a = 1,
+Qy = 0,
+where y is a column vector of size DL with y
+=
+(y1
+1, · · · , yL
+1 , · · · , y1
+D, · · · , yL
+D)T as its components; D is an
+5We note that the problem can also solved with the same approach
+(but heavier notation) by more generally defining ωk(A[t], u(A[t])) to be
+functions of both the age and the action.
+
+upper bound on the age state in the system which can be
+set sufficiently large so that the probability of reaching D
+is vanishing.6 Qy = 0 is the matrix representation of the
+following (global balance) equations:
+L
+�
+l=0
+yl
+a+1 −
+L
+�
+l=0
+yl
+a(1 − µ)l = 0
+∀a = 1, · · · , D − 2,
+L
+�
+l=0
+�
+1 − (1 − µ)l�
+yl
+D −
+L
+�
+l=0
+yl
+D−1(1 − µ)l = 0,
+−
+L
+�
+l=0
+yl
+1(1 − µ)l +
+D
+�
+a=2
+L
+�
+l=0
+yl
+a
+�
+1 − (1 − µ)l�
+= 0.
+If this LP is feasible, and y is an optimal solution, then the
+optimal policy u∗(a) is a probabilistic policy, whereby the
+probability f l
+a of choosing l channels when the age is at state
+a equals:
+f l
+a =
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+yl
+a
+L
+�
+l=0
+yl
+a
+,
+if
+L
+�
+l=0
+yl
+a ̸= 0
+1
+L,
+if
+�
+l
+yl
+a = 0
+(3)
+for l = 0, 1, · · · , L and a = 1, 2, · · · , D.
+Proof:
+As shown in [25], it is enough for us to optimize
+over the Markovian policies for Problem 1. Since the process
+is not affected by a shift in time, we can define the probabilistic
+scheduling policy where f l
+a denotes the probability of choosing
+l channels when the AoI of single source is at state a. The
+normalization constraint of the probabilistic scheduling policy
+requires
+L
+�
+l=0
+f l
+a = 1 and f l
+a ⩾ 0 for all a.
+Notice that the system state can be fully characterized by a
+one-dimensional Markov chain with age A[t] as state. Given
+the current state information A[t], the system state at the next
+time slot A[t+1] depends only on the current state A[t] (with
+no dependence on earlier states) and the current action u[t].
+In addition, the objective and constraints only depend on the
+current state and action. So an equivalent MDP problem can
+be formulated. Let λa2
+a1 denote the transition probability from
+state a1 to a2, and define ¯µ ≜ 1 − µ as the probability of
+channel failure. Then based on the channel success model,
+λa2
+a1 =
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+L
+�
+l=1
+f l
+a1 ¯µl,
+1 ≤ a1 ≤ D − 1, a2 = a1 + 1
+L
+�
+l=1
+f l
+a(1 − ¯µl),
+a1 = 1, · · · , D, a2 = 1
+L
+�
+l=1
+f l
+D(1 − ¯µl),
+a1 = D, a2 = D
+0,
+otherwise.
+(4)
+Since there are finitely many states, there exists a stationary
+distribution π(a) for every a. Let C be the set of all recurrent
+6In practice, moderate level of D is enough so that the dimension of LP
+won’t be large. Also, when there is only age violation related objective and
+constraints, it’s enough to set D = d + 1. See III-C and IV-C for references.
+states, then C is irreducible and closed, thus C is positive
+recurrent. When a ∈ C the stationary distribution π(a) is equal
+to the long term average lim
+T →∞
+1
+T
+T
+�
+t=1
+1{A[t] = a} independent
+of the starting point. When state a /∈ C, then both the stationary
+distribution and the long term average are equal to zero. So the
+optimization problem is equivalent to the following constraint
+MDP problem:
+min
+f la
+D
+�
+a=1
+π(a)ω0(a)
+s.t:
+D
+�
+a=1
+L
+�
+l=0
+π(a)f l
+al ≤ b
+D
+�
+a=1
+π(a)ωk(a) ≤ ck, k = 1, · · · , K
+(5)
+L
+�
+l=0
+f l
+a = 1, f l
+a ⩾ 0
+∀a ≤ D, l ≤ L
+(6)
+H · Π = Π,
+1 · Π = 1
+(7)
+where Π = [π(1), · · · , π(D)]T is the stationary distribution of
+the Markov Chain and H is the D × D transition matrix with
+hij = λi
+j. Let us define yl
+a = π(a)f l
+a, then π(a) =
+L
+�
+l=0
+yl
+a for
+a ≤ D. Then the constraint 5 becomes:
+D
+�
+a=1
+L
+�
+l=0
+yl
+aωk(a) ≤ ck, k = 1, · · · , K.
+The
+normalization
+constraint
+in
+Equation
+7
+requires
+D
+�
+a=1
+L
+�
+l=0
+yl
+a = 1. Substituting yl
+a into the CMDP problem and
+after simplifying, we establish the equivalency of the Linear
+Programming problem. After obtaining the solution y, we
+let f l
+a = yl
+a/π(a) for π(a) ̸= 0.States a with π(a) = 0, are
+transient states, and the actions at these states do not affect
+the average results. For those states we adopt a simple policy
+as in Equation 3, then the constraint 7 is also satisfied.
+C. Characterization and Insights on Age-Optimal Schedulers
+Our general framework encompasses a wide range of objec-
+tives and constraints for different choices of ωk(·) functions
+using different age and age-violation metrics. In this section,
+we focus on two important problems that can be expressed
+within our framework: average age minimization and age-
+violation-rate minimization. This effort will enable us to
+characterize their optimal schedulers and gain insights into
+their nature.
+Optimal scheduler minimizing average age: When we set
+ω0(a) = a in (1), the objective of the optimization problem
+becomes to minimize the average age
+lim
+T →∞
+1
+T
+T
+�
+t=1
+E{A[t]} =
+D
+�
+a=1
+a π(a).
+
+0
+5
+10
+15
+AoI
+0
+2
+4
+6
+8
+Average number of activated channels
+=0.12
+=0.1
+=0.08
+=0.04
+Figure 2. Optimal number of channels to choose to minimize average AoI
+when b = 2.
+For this problem formulation, we retain the energy constraint
+lim
+T →∞
+1
+T
+T
+�
+t=1
+E [u (A[t])] ≤ b; but do not need additional age
+constraints. Hence, ωk(a) = 0 and ck = 0, for all k and a.
+Figure 2 depicts the average number of activated channels
+of the average-age optimal scheduler as a function of the age
+states under different channel success probabilities µ for the
+energy constraint b = 2. We will further discuss these results
+at the end of this section in comparison with the next scheduler
+of interest.
+Optimal scheduler minimizing age-violation-rate: Setting
+ω0(a) = 1{a > τ} n (1), the objective becomes minimizing
+the average age-violation-rate
+lim
+T →∞
+1
+T
+T
+�
+t=1
+E{1{A[t] > τ}} =
+D
+�
+a=τ+1
+π(a).
+As before, we keep the energy constraint, but do not need
+additional age constraints. Hence, ωk(a) = 0 and ck = 0, for
+all k and a.
+With this, the problem becomes minimizing the age-
+violation-rate under an energy constraint. Unlike in the previ-
+ous problem, our goal is not to minimize the average age but
+to avoid age-violation events. In this scenario, we can view
+all the states with a > τ as state τ + 1, so it’s enough to set
+D = τ + 1.
+Figure 3 depicts the average number of activated channels
+of the violation-rate optimal scheduler as a function of the age
+states under different channel success probabilities µ for age
+threshold τ = 8 and the same energy constraint b = 2. Next,
+we compare the optimal policies of these two schedulers and
+discuss the insights that can be gained from their study.
+Insights on the two optimal schedulers: We start by noting
+the similarities of the optimal policy under both scenarios:
+(i) Each optimal policy is a probabilistic combination of at
+most two deterministic policies, which matches the result
+that the number of randomization is at most the number
+of constraints, as shown in [25].
+(ii) For each scenario, as the channel success probability
+increases, the corresponding optimal policy starts trans-
+2
+4
+6
+8
+AoI
+0
+5
+10
+15
+20
+25
+Average number of activated channels
+=0.12
+=0.1
+=0.08
+=0.04
+Figure 3. Optimal number of channels to choose to minimize AoI violation
+rate when b = 2 and τ = 8.
+mitting at lower age levels, and also tends to choose
+more channels at the same age level. This is a somewhat
+counter-intuitive characteristic that indicates that the opti-
+mal policy should be more active and active earlier when
+the channels are more reliable.
+(iii) The optimal policy in each scenario is idle when AoI is
+relatively small. This is meaningful once we observe that,
+when the age is relatively small, a successful transmission
+will not benefit the objective as much as when the age
+is large. Hence, the optimal scheduler saves energy for
+larger age states.
+However, we also notice differences between the two sets
+of schedulers:
+(i) The optimal policy in the average age minimization prob-
+lem has an activation function u∗(·) that is monotone non-
+decreasing with increasing age state. On the other hand,
+the monotonicity does not hold in the age violation rate
+minimization problem. This difference comes from the
+non-convex nature of the the age violation rate function
+in the latter case. In [25] and many related works (e.g.,
+[9], [34]), the authors exploit the monotone structure and
+threshold nature of the optimal scheduling policy for solv-
+ing the CMDP, revealing insights as well as simplifying
+the algorithm by using the convexity or concavity of the
+objective functions. However, in our general treatment,
+the objective functions, such as age violation rate, are not
+necessarily convex or concave, which prevents us from
+using the same approach. Hence, to obtain the optimal
+policy, we use the generally applicable LP method despite
+the higher computational complexity that it may require
+in order to develop insights about the optimal solution.
+(ii) In the average age minimization problem, the number
+of activated channels of the optimal policy experiences
+a sub-linear/concave like increase with respect to ages
+after the age level that the number of activated channels
+starts to be above zero. In contrast, the age violation rate
+minimizing schedulers experience a super-linear/convex
+like increasing with respect to age until the deadline level
+τ. This difference can be interpreted as follows: in the age
+
+1
+2
+3
+4
+5
+6
+AoI
+0
+5
+10
+15
+20
+25
+Average number of activated channels
+=0.1
+=0.001
+=0.0001
+Figure 4.
+Optimal number of channels to choose to minimize average age
+under violation rate constraint when τ = 5, b = 3, µ = 0.2
+violation rate minimization problem, the penalty happens
+only when the age is beyond the age deadline, and hence
+the optimal scheduler will be more aggressive as the
+threshold level is approached from below. In contrast,
+for the average age minimization problem, the number
+of activated channels increases more gradually to balance
+the tradeoff between consuming energy unnecessarily at
+very low age levels and waiting too long to consume the
+available energy, which yields an indefinitely increasing
+cost.
+These insights on the structure of the allocation functions of
+the optimal schedulers can guide designers in restricting their
+search to classes of functions with sufficiently flexible but also
+tractable forms whenever the solution through the LP strategy
+is not possible due to lack of prior statistical information as
+well as computational resources.
+To demonstrate how the age violation rate constraint effects
+the shape of the scheduler more clearly, in Figure 4 we set
+the objective function to be ω0(a) = a, the energy constraint
+to be b = 3, and the channel success probability to be µ =
+0.2. In addition, we set ω1(a) = 1{a > τ}, where the age
+deadline τ = 5. We set c1 = ϵ and show how the number of
+activated channels changes over age states under different ϵ
+levels. By adding and tightening the tolerance constraint, we
+can see the transition from concave (or sublinear) to convex
+(or superlinear) form. As such, the optimal scheduler becomes
+more aggressive when the age increases. This reveals a trade-
+off between the average age and the age-violation-rate, namely
+that reducing the age violation rate calls for an increasingly
+more aggressive allocation function.
+IV. AGE-OPTIMAL MULTI-CHANNEL SCHEDULING FOR
+MULTIPLE USERS
+In this section, we extend our framework to the general
+multi-user multi-channel age-optimal scheduling problem. As
+before, this formulation allows us to cover a range of scenarios
+depending on the choice for objective function and constraints.
+To that end, we investigate the feasibility and stability region
+of the optimal policy along with alternatives from related
+literature associated with multi-user settings.
+A. Problem Formulation
+The formulation of the optimization problem for the multi-
+user case is similar to single user case (1):
+min
+u(A)
+lim
+T →∞
+1
+T
+T
+�
+t=1
+E [ω0(A[t])]
+(8)
+s.t :
+lim
+T →∞
+1
+T
+T
+�
+t=1
+E [ui (A[t])] ≤ bi, i = 1, · · · , n,
+lim
+T →∞
+1
+T
+T
+�
+t=1
+E [ωk (A[t])] ≤ ck, k = 1, · · · , K,
+ui(A[t]) ∈ {0, 1, · · · , L}, i = 1, · · · , n,
+n
+�
+i=1
+ui(A[t]) ≤ L
+where = (u1(A), · · · , un(A)) denotes the scheduling policy
+at state A with ui(A) as the number of channels allocated to
+source i. The weight functions ωk(·), k = 0, 1, · · · , K, map
+the age states to cost values that capture age-related objectives
+and constraints. Source nodes can have heterogeneous energy
+constraints bi, which means node i can transmit over at most
+bi channels per slot on average.
+B. Performance analysis
+Next, we establish the equivalence of the multi-user problem
+formulation to a linear programming (LP) problem, as we
+did for the single user case in Section III-B. To enable a
+more compact notation, we will use a ≜ (a1, a2, · · · , an)
+and l ≜ (l1, l2, · · · , ln) to denote values of A[t] and u(A),
+respectively. We further define sets A ≜ {1, · · · , D}n, L ≜
+{1, · · · , L}n, and L1 ≜ {l : lΣ ≤ L} where lΣ ≜
+n
+�
+i=1
+li.
+Theorem 2: The solution of the multi-user age-optimization
+problem (8) can be obtained by solving the following linear
+programming problem:
+min
+yl
+a
+�
+a∈A
+�
+l∈L1
+yl
+aω0(a)
+s.t:
+�
+a∈A
+�
+l∈L1
+yl
+ali ≤ bi, i = 1, 2, · · · , n
+0 ≤ yl
+a ≤ 1
+∀l ∈ L, a ∈ A
+yl
+a = 0
+∀l ∈ L/L1
+�
+a∈A
+�
+l∈L1
+yl
+a = 1
+�
+a∈A
+�
+l∈L1
+yl
+aωk(a) ≤ ck, k = 1, · · · , K
+(9)
+Qy = 0
+where y is a column vector with yl
+a as components and
+Q represents the transition matrix associated with the age
+
+dynamics, exactly in the same form as in the single-user case
+(cf. Theorem 1).
+If this LP is feasible and y is an optimal solution, then
+the optimal policy u∗
+i (a) is a probabilistic policy, whereby
+the probability f l
+a of choosing l channels for source nodes
+i = 1, · · · , n when the AoI is at state a equals:
+f l
+a =
+�
+�
+�
+�
+�
+�
+�
+�
+�
+yl
+a
+�
+l∈L
+yl
+a
+,
+if
+�
+l∈L
+yl
+a ̸= 0
+1
+|L|,
+if
+�
+l∈L
+yl
+a = 0
+for l ∈ L and a ∈ A.
+Proof:
+We will use f l
+a to denote the probability of choosing
+l = (l1, · · · , ln) channels for source nodes (1, · · · , n) when
+the AoI is at state a. Thus �
+l∈L
+f l
+a = 1, and f l
+a ≥ 0 for all a.
+Similarly as in Theorem 1, the constraint MDP problem with
+n−dimensional Markov Chains for multi-user scheduling can
+be generally formulated as:
+min
+�
+a
+π(a)ω0(a)
+s.t:
+�
+a
+�
+l
+π(a)f l
+ali ≤ bi, i = 1, 2, · · · , n
+(10)
+f l
+a = 0
+∀l ∈ L/L1
+�
+a
+π(a)ωk(a) ≤ ck
+k = 1, · · · , K
+H · Π = Π,
+1 · Π = 1,
+(11)
+where the indices range over a ∈ A and l ∈ L; π(a) is the
+stationary distribution of state a; and ωk(a), k = 0, 1, · · · , K,
+are age related objective and cost functions. The constraints 10
+bound the average energy of nodes i by bi for i = 1, · · · , n.
+In the constraint 11, Π is a Dn × 1 stationary distribution
+vector with π(a), a
+∈
+A as entries.7 H represents the
+Dn×Dn transaction matrix with hi,j equals the probability of
+transaction from the jth state in Π to the ith state in Π, which
+can be detailed by using the age evolution and channel success
+probability equations similarly as in Equation 4. Similarly, we
+will define
+yl
+a ≜ yl1,l2,··· ,ln
+a1,a2,··· ,an = π(a)f l
+a.
+By changing the value of the weight functions, we can get
+different AoI related metrics, but all are linear with respect to
+yl
+a. Then,
+π(a) =
+�
+l
+yl
+a,
+and the normalization constraint requires:
+�
+a
+�
+l
+yl
+a = 1.
+Substituting yl
+a into the CMDP problem, we obtain the equiv-
+alence of the LP problem.
+7The existence of the stationary distribution follows by the same proof as
+in Theorem 1.
+C. Characterization and insights on multi-user scheduling
+problem with violation tolerance Constraints
+Since there is no closed form solution to the general age-
+optimal problem, we will study the multi-user single-channel
+scheduling feasibility problem with age-violation tolerance
+constraint as a common setting to investigate its performance
+and characteristics.
+In particular, we will compare the stability region of the
+optimal scheduler with a previously developed algorithm that
+was developed for the special case of multi-user single-channel
+setting [21]. To that end, we set L = 1 and bi > 1. Thus, all
+the energy constraints will be inactive, and we can focus on the
+tolerance constraint, as in [21]. Since we are only interested
+in feasibility, we set ω0(a) = 1 for all a. To express the
+age-violation rate constraints we define the weight functions
+ωk(a) =
+�
+0,
+if ak ≤ τk
+1,
+if ak ≥ τk + 1,
+and set ck = ϵk for k = 1, 2, · · · , K = n, to represent the
+heterogeneous age-violation tolerance level for the kth source.
+Then the constraint
+�
+a
+π(a)ωk(a) ≤ ck becomes
+πk(τk + 1) ≤ ϵk
+∀k = 1, · · · , K = n,
+where πk(τk + 1) denotes the total probability (under the
+stationary distribution) that source k violates its age threshold
+τk. Since
+πk(τk+1) =
+�
+j1,...,jk−1,jk+1,...,jn
+π(j1, ...jk−1, τk+1, jk+1..., jn),
+the constraint (9) in the linear programming problem becomes
+�
+j1,...,jk−1,jk+1,...,jn
+�
+l
+yl
+j1,...,jk−1,τk+1,jk+1,...,jn ≤ ϵk.
+For the sake of easy visualization, we study the case with
+n = 2 users. In this case, the LP problem is formulated as:
+min
+1
+s.t:
+0 ≤ yl1,l2
+a1,a2 ≤ 1
+∀l1, l2 = 0, 1
+yl1,l2
+a1,a2 = 0
+∀l1 + l2 > 1
+�
+j
+�
+l1,l2
+yl1,l2
+τ1+1,j ≤ ϵ1
+�
+j
+�
+l1,l2
+yl1,l2
+j,τ2+1 ≤ ϵ2
+The numerical results can be seen in Figures5 and 6 for
+different parameters where the upper right area of the solid
+blue line is the stability region of the optimal scheduler.
+These typical examples reveal the non-negligible gap between
+the performance of the optimal scheduler and the previously
+proposed design, even for a small two user setting.
+This motivates the search for new algorithms that can
+perform closer to the optimal scheduler, even when the channel
+statistics are unknown a priori. This is performed in the next
+section along with further discussion about these numerical
+results after we discuss our online scheduling algorithm.
+
+Before we proceed, we note even the above numerical
+results are for two-user single-channel scheduling problem
+under tolerance constraints for visualization purposes, our
+methods apply to the more general multi-user multi-channel
+scheduling problem under violation tolerance and energy
+constraints. Although the computational complexity may be
+relatively high for the LP solution compared to other solutions
+that exploit the special structure of particular problems, as we
+mentioned above, due to the non-convexity and non-concavity
+of the tolerance constraints, the monotone and threshold
+structure of the optimal policy does not hold. The Whittle
+Index approach (used, for example, in [29], [31]) which have
+relatively low complexity also does not apply to our multi-
+channel scheduling problems since each user in our setting
+is allowed to transmit over multiple channels simultaneously,
+whereby the Whittle’s Indexability condition does not hold.
+Using the generally applicable LP-based approach reveals key
+insights that can guide the designers in developing efficient
+schedulers for future multi-channel wireless technologies.
+V. ONLINE SCHEDULING UNDER UNKNOWN CHANNEL
+STATISTICS
+Until this point, we have assumed that the channel success
+probabilities are known when solving the optimization prob-
+lems. In this section, we use a Lyapunov-drift-plus-penalty
+approach(see [28]) to solve the multi-user online age related
+optimization problem in the scenario when only the current
+channel states are known, but the channel statistics are un-
+known.
+We will transfer all the energy and age-related constraints
+into the virtual queues and view the objective as a penalty term
+with parameter M. For the energy constraint of the source i,
+let us define the corresponding virtual queue as Q1,i[t], whose
+initial value is Q1,i[0] = 0 and update equation is:
+Q1,i[t + 1] = (Q1,i[t] + ui (A[t]) − bi)+ .
+Similarly, we define the virtual queue Q2,k[t] for the kth age-
+related constraint, whose initial value is Q2,k[0] = 0 and
+update equation is:
+Q2,k[t + 1] = (Q2,k[t] + ωk (A[t]) − ck)+ .
+Generically, if the virtual queue Q1,i[t] is stable, then its input
+rate lim
+T →∞
+1
+T
+T
+�
+t=1
+E [ui (A[t])] will be less than its output rate
+bi [28], so that the corresponding constraint can be satisfied.
+Define the state of both virtual queues and age at time t
+as Q[t] = (Q1,1[t], · · · , Q1,n[t], Q2,1[t], · · · , Q2,K[t], A[t]).
+Based on the virtual queues, we will define the quadratic
+Lyapunov function as:
+V [t] = 1
+2(
+n
+�
+i=1
+Q2
+1,i[t] +
+K
+�
+k=1
+Q2
+2,k[t]),
+and develop an online algorithm to greedily minimize
+the upper bound of the Lyapunov-drift-plus-penalty func-
+tion ∆V (q) + ME[ω0(a)] given the current state q
+=
+(q1,1, · · · , q1,n, q2,1, · · · , q2,K, a), where:
+∆V (q) = E[V [t] − V [t − 1]|Q[t] = q].
+We consider the multi-user single-channel scheduling prob-
+lem under tolerance constraints as a specific example to
+present the design. Since there are no energy constraints,
+we do not need the set of virtual queues {Q1,i[t]}i. In
+order to express the kth violation rate constraint for source
+k = 1, · · · , n, we let ωk (A[t]) = 1 (Ak[t + 1] > τk) and
+ck = ϵk. Then the virtual queue Q2,k[t], whose initial value
+is Q2,k[t] = 0, updates as follows:
+Q2,k[t + 1] = (Q2,k[t] + 1 (Ak[t + 1] > τk) − ϵk)+ ,
+where Ak[t + 1] = 1 + Ak[t](1 − Sk[t]Uk[t]); Sk[t] represents
+the channel success; Uk[t] represents whether the source is
+scheduled to transmit or not. If virtual queue Q2,k[t] is stable,
+its input rate, the threshold violation rate πk(τk + 1) =
+limT →∞ 1
+T
+�T
+t=1 1 (Ak[t + 1] > τk) , will be less than its
+output rate ϵk.
+The conditional Lyapunov drift can be bounded as follows:
+∆V (q)
+≤
+n
+�
+k=1
+q2,kE [Rk − ϵk|q2,k] +
+n
+�
+k=1
+E
+�
+(Rk − ϵk)2
+2
+|q2,k
+�
+,
+where Rk
+∆= 1{1 + Ak (1 − SkCk) > τk}. At every time slot
+t, we can develop an online algorithm as summarized below
+to greedily minimize the upper bound of the Lyapunov drift
+given the queue lengths Q[t − 1] and A[t − 1] since there is
+no objective or penalty term in this case.
+Algorithm 1 A Heuristic Scheduling Policy
+1: Input current system state: Ai[t],Qi[t].
+2: Define available transmission decision set: only one Ui[t]
+can be 1.
+3: Choose U[t] to minimize the upper bound of Lyapunov
+drift function in the above inequality.
+4: Update queue lengths for next time slot.
+Again, for the sake of easy visualization, we will only
+present the simulation results for the two-user online schedul-
+ing problem under age tolerance constraints, but the online
+algorithm can be simply applied to any number of sources.
+The simulation results are illustrated in Fig 5 and Fig 6 for
+different parameters where the upper right area of the dash-dot
+purple line is the stability region of the online scheduler when
+the channel condition µi. The comparison will be in the next
+section.
+VI. COMPARISON OF STABILITY REGIONS UNDER AGE
+VIOLATION CONSTRAINTS
+In this section, we compare the performance of three dif-
+ferent algorithms for the two-user single channel scheduling
+feasibility problem under age violation tolerance constraints.
+These are: the optimal scheduler from Section IV; the prior
+
+design from [21] developed for a single-channel multi-user
+setting; and our online scheduler from Section V that does
+not require channel statistics.
+We first focus on the case when the two source nodes are
+symmetric. In Figure 5, there are two source nodes with the
+same age thresholds of τ1 = τ2 = 2 and the same channel
+success probabilities of µ1 = µ2 = 0.85. The upper right
+area of the blue line is the stability region for the optimal
+scheduling algorithm in Section IV-C. The yellow and orange
+lines correspond to the algorithm in [21] and capture the two
+cases when the rate vector does or does not possess a special
+property (called step-down rate vector). The purple line marks
+the stability region for the online algorithm when the channel
+conditions µ1, µ2 are unknown. Several observations are in
+order from these simulation results:
+(i) The stability regions are all symmetric, as can be expected
+due to the homogeneous deadline thresholds and channel
+conditions.
+(ii) The optimum policy (blue line) outperforms other poli-
+cies, with markedly better performance in cases where
+the tolerance levels are greatly different from each other.
+(iii) The online algorithm (purple line) performs very closely
+to the optimal policy, experiencing a small performance
+loss only at some extreme range of tolerance levels.
+(iv) When compared with the algorithms from [21](yellow
+and red lines), the online algorithm performs particularly
+better when one of the tolerance rates is smaller than the
+corresponding channel loss probability, as observed by
+the vertical gap between purple and yellow lines.
+(v) The online and optimal policies are continuous with
+respect to the tolerance level, which eliminates the need
+to check if the tolerance rate vector satisfies certain
+properties, such as the step-down rate condition in [21].
+To compare the advantages and disadvantages of the al-
+gorithms under non-homogeneous scenarios, in Figure 6, we
+consider two source nodes with asymmetric age thresholds of
+τ1 = 2, τ2 = 4 and a common channel success probability of
+µ1 = µ2 = 0.85. Since the violation rate depends on both
+the age thresholds and the channel success probabilities, this
+is a non-homogeneous scenario even though µ1 = µ2. In this
+figure, in contrast to the previous figure, we can further see
+that the optimal policy outperforms others when one of the
+0
+0.2
+0.4
+0.6
+0.8
+1
+1
+0
+0.2
+0.4
+0.6
+0.8
+1
+2
+Optimal scheduling algrithm
+Algorithm proposed in [21]
+under general rate vector
+Algorithm proposed in [21]
+under step-down rate vector
+Online scheduling algorithm
+Figure 5. Stability region (upper-righter) comparison for symmetric case.
+0
+0.2
+0.4
+0.6
+0.8
+1
+1
+0
+0.2
+0.4
+0.6
+0.8
+1
+2
+Optimal scheduling algrithm
+Algorithm proposed in [21]
+under general rate vector
+Algorithm proposed in [21]
+under step-down rate vector
+Online scheduling algorithm
+Figure 6. Stability region (upper-righter) comparison for asymmetric case.
+tolerance constraints is very strict, namely when ϵ1 approaches
+1. In this regime, the feasible tolerance level ϵ2 of user 2
+other algorithms is bounded away from zero while the optimal
+algorithm decreases towards zero.
+These simulation results are typical of other circumstances,
+with the common observation that our online scheduler per-
+forms close to the optimal scheduler and typically non-
+negligibly better than the most closely related state-of-art
+algorithm from [21], despite the fact that it operates without
+the knowledge of channel statistics that is assumed in the other
+designs.
+VII. CONCLUSIONS
+In this paper, we considered a general class of age-optimal
+scheduling problems for multi-source multi-channel commu-
+nication. We formulated the generic age-optimization problem
+with flexible weight functions ωk under energy and tolerance
+constraints in the form of a CMDP. We solved this generic
+problem, which a usual threshold-based structure policy does
+not apply, by relating it to the solution an associated linear
+programming problem using the powerful theory of CMDPs.
+Then, we focused on the special case of single-source multi-
+channel scenario to investigate the characteristics of optimal
+scheduler for the important special cases of average-age and
+violation-rate minimization.
+Our investigations revealed several interesting insights, in-
+cluding the observation that age-violation-rate minimizing
+scheduler employs a super-linearly like growing energy al-
+location strategy with increasing age, as opposed to the
+sub-linearly like growing allocation for the average-age-
+minimizing scheduler. These insights may provide useful
+guidelines for IoT network designers in developing effective
+update strategies based on different sensitivities of applications
+to age performance.
+We also studied the special case of multi-source single-
+channel scheduling problem with age violation rate constraints
+to investigate the feasibility region of the optimal scheduler
+together with that of most closely related prior works. Finally,
+we have developed an online scheduler that does not require
+the knowledge of channel statistics, and compared its perfor-
+mance to the optimal scheduler through simulations to observe
+that it performs closely to the optimal scheduler despite its lack
+of information on channel statistics.
+
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+Higher-Order Patterns Reveal Causal Timescales
+of Complex Systems
+Luka V. Petrovi´c1, Anatol Wegner2, and Ingo Scholtes1,2
+1Data Analytics Group, University of Zurich, Z¨urich, Switzerland
+2Center for Artificial Intelligence and Data Science (CAIDAS),
+Julius-Maximilians-Universit¨at W¨urzburg, W¨urzburg, Germany
+January 30, 2023
+Abstract
+The analysis of temporal networks heavily depends on the analysis of time-respecting paths.
+However, before being able to model and analyze the time-respecting paths, we have to infer
+the timescales at which the temporal edges influence each other. In this work we introduce
+temporal path entropy, an information theoretic measure of temporal networks, with the aim to
+detect the timescales at which the causal influences occur in temporal networks. The measure
+can be used on temporal networks as a whole, or separately for each node. We find that the
+temporal path entropy has a non-trivial dependency on the causal timescales of synthetic and
+empirical temporal networks.
+Furthermore, we notice in both synthetic and empirical data
+that the temporal path entropy tends to decrease at timescales that correspond to the causal
+interactions. Our results imply that timescales relevant for the dynamics of complex systems
+can be detected in the temporal networks themselves, by measuring temporal path entropy.
+This is crucial for the analysis of temporal networks where inherent timescales are unavailable
+and hard to measure.
+1
+Introduction
+The research of dynamic complex systems has in recent years advanced beyond static graph repre-
+sentations [Lambiotte et al., 2019, Battiston et al., 2020]. The focus has shifted to various general-
+izations of diadic interactions in graphs: multiple types of interactions in multilayer network [Kivel¨a
+et al., 2014], multibody interactions in the form of simplicial complexes and hypergraphs [Petri and
+Barrat, 2018] and models that incorporate concepts of memory [Scholtes et al., 2014, Lambiotte
+et al., 2015, Williams et al., 2022]. Such generalized relationships allow us to better model complex
+systems, because they can represent richer data.
+Temporal networks are one kind of such rich data; they record not only who interacted with
+whom, but also when each interaction happened. They bring us closer to understanding the dynamics
+of complex systems, but require us to perform analysis beyond the static networks approach [Holme
+and Saram¨aki, 2012, Holme, 2015]. The time information can yield valuable insights on its own [Goh
+and Barab´asi, 2008], and, although initially the temporal and topological aspects of temporal net-
+works were mostly studied independently, even richer insights are hidden in the coupling of the
+temporal and topological patterns [Ceria et al., 2022]. Such coupling can affect the statistics of
+1
+arXiv:2301.11623v1  [physics.soc-ph]  27 Jan 2023
+
+time-respecting paths [Holme and Saram¨aki, 2012] in temporal networks, which impacts e.g., anal-
+ysis of accessibility [Lentz et al., 2013], reachability [Badie-Modiri et al., 2020], spreading [Masuda
+et al., 2013, Scholtes et al., 2014, Lambiotte et al., 2015, Badie-Modiri et al., 2022], clustering [Ros-
+vall et al., 2014], centralities [Scholtes et al., 2016], and visualization [Perri and Scholtes, 2020].
+Although there are many possible ways in which temporal and topological patterns can couple in
+complex systems, one of the most basic cases is when an incoming temporal edge to a node causes
+a change of frequencies of the edges emanating from the node within a given time-window. For
+instance, in a communication network we expect an incoming message to induce a outgoing message
+on the same topic, e.g. in the form of a reply, within a certain time window reflecting the minimal
+reaction time and memory of the recipient. However, information on the timescales relevant for the
+temporal network dynamics is rarely available in real world settings.
+In this work, we define an information theoretic measure to detect timescales at which interactions
+in a complex system cause each other. We demonstrate its effectiveness in both synthetic and real
+world data.
+2
+Background
+Let Γ = (V, E) be a temporal network consisting of a set of nodes V and a set of time-stamped edges
+E ⊆ V × V × R. We denote the set of unique edges with E ⊂ V × V . A temporal edge (v, w, t) ∈ E
+represents a direct link from node v to node w at time t. For simplicity, we assume that the temporal
+edges are instantaneous, however the method and algorithms can be modified in a straightforward
+fashion to the case where edges have finite duration. Formally, we call a sequence of time-stamped
+edges (v1, w1, t1), . . . , (vk, wk, tk) a time-respecting path iff for all i ∈ {2, . . . , k} they satisfy the
+following conditions [Pan and Saram¨aki, 2011, Holme and Saram¨aki, 2012, Casteigts et al., 2021]:
+wi−1 = vi
+(1)
+τmin < ti − ti−1 < τmax.
+(2)
+The parameters τmin and τmax naturally introduce a timescale that affects all analyses of temporal
+networks that are based on time-respecting paths.
+The timescale has to be defined differently for processes on the temporal network or the processes
+of the temporal network [Holme and Saram¨aki, 2012]. In the former case, the timescale is defined by
+the process running on the temporal network, e.g. in the case of an epidemic that is spreading over
+a temporal network of contacts, the timescale is a property of a disease, related to the time interval
+in which a person is contagious and not related to the timescales at which contacts occur 1. In the
+latter case, the timescale is part of the process of edge activation, and thus shapes the temporal
+network itself. For example, information that is spreading between individuals is also affecting the
+individuals’ choice with whom to share the information: a person would be more likely to share the
+family-related information with a family member and work-related information with a colleague. In
+this letter, we investigate the latter case, more specifically, we investigate whether interactions in a
+complex system induce one another at a given timescale τ = [τmin, τmax].
+In the literature, there exist a variety of definitions of timescales in temporal networks, as well as
+a variety of methods aimed at detecting them. The various definitions of timescales are based on the
+different structural features of temporal networks. One popular definition of timescales in temporal
+networks is the approach based on splitting the network into time-slices and aggregating the edges
+1We note that the processes on and of the temporal network may interact [Gross and Sayama, 2009], and thus blur
+the distinction.
+2
+
+inside the time-interval [Caceres and Berger-Wolf, 2013, Darst et al., 2016]. In the same framework,
+Ghasemian et al. [2016] and Taylor et al. [2016] investigate the limitations of detectability of cluster
+structures dependent on the timescales of aggregation. Since this framework is based on aggregating
+the temporal network into a sequence of static time-aggregated networks, it loses information of
+the time-respecting paths and is therefore not in line with our aims. Other lines of research often
+related to timescale detection are change point detection [Peixoto and Gauvin, 2018], and analysis
+of large-scale structures. Gauvin et al. [2014] detects clusters and their temporal activations in a
+temporal network using tensor decomposition. Similarly, Peixoto [2015] proposed a method to detect
+the change points of cluster structure in a temporal network. Peixoto and Rosvall [2017] proposed
+a method to simultaneously detect the clusters and timescales in temporal network, however, they
+model the temporal network as a single sequence of tokens (similar to [Peixoto and Gauvin, 2018])
+that represent temporal edges, and their timescale inference refers to the number of tokens in the
+memory of a Markov chain that models such a sequence. In our view, these works focus on mesoscale
+structures, and take a coarse grained view of temporal networks, while in this work, we propose a
+complementary approach by focusing on local correlations between temporal edges incident on a
+node and subsequent temporal edges emanating from it. Among the works that took a fine-grained
+view, Williams et al. [2022] investigated pairwise correlations between the temporal edges. Different
+from the approach that we took, they aggregate the network in time-slices as a preprocessing step,
+and the timescale is defined as a maximum number of time slices back in time at which correlations
+are detectable. Scholtes et al. [2016] found that correlations between edges on time-respecting paths
+affect centralities; they modeled the time-respecting paths with higher-order models and found that
+this approach improves the centrality rankings. They identified the issue of timescale detection in
+the context of time-respecting paths, which our work addresses. Our work also complements the
+work of Pfitzner et al. [2013] which introduces betweenness preference that can be used to study
+over- and under-represented time-respected paths in temporal networks, but does not address the
+problem of detecting the timescales at which these paths occur. To the best of our knowledge, our
+work is the first to address the issue of timescale detection for time-respecting paths in temporal
+networks.
+3
+Temporal Path Entropy
+We address the issue of timescale detection by analysing the statistics of time-respecting paths
+Pk
+τ (Γ) of length k at timescales τ = [τmin, τmax] in a temporal network Γ. We define “temporal path
+entropy” H for paths (v0, v1, . . . , vk) of length k as the entropy of the last node vk conditional on
+the sub-path (v0, v1, . . . , vk−1):
+H = H(vk|v0, . . . , vk−1)
+(3)
+= H(v0, . . . , vk) − H(v0, . . . , vk−1),
+(4)
+where H(P) = − �
+i pi ln(pi) is the Shannon entropy.
+The identity in Eq. 4 can be obtained
+by applying the chain rule (see Appendix for derivation). By definition, temporal path entropy
+H measures uncertainty in the last step of time-respecting paths given the k − 1 previous steps.
+A lower value of the entropy indicates a high correlation between the memory of time-respecting
+paths and subsequent steps. Hence the τ for which the entropy reaches its minimum gives us the
+timescale for which time-respecting paths become most predictable, i.e.
+where the correlations
+between subsequent temporal edges are the most pronounced. The entropy can also be defined for
+a single node v, by simply fixing vk−1 = v, allowing for a more fine grained analysis that could be
+important if nodes differ significantly with respect to the timescales they operate on. The intuition
+3
+
+behind the temporal path entropy is to measure how much the target vk of an edge emanating
+form the node vk−1 depends on the incoming paths that influenced it in the past. Testing those
+dependencies at different timescales would thus point to the timescales at which the dependencies
+are most pronounced. When we compute temporal path entropy for the whole temporal network,
+we use all time-respecting paths of length k in the temporal network, while when we compute it for
+a node v, we select only the paths where vk−1 = v.
+1.75
+2.00
+2.25
+H[ nat ]
+Synthetic-2
+0
+1
+2
+WB-DE
+2
+3
+HC-email
+50
+100
+150
+200
+250
+0.0
+0.5
+1.0
+counts [103]
+100
+102
+104
+106
+time [s]
+0
+20
+103
+105
+0.00
+0.05
+0.10
+m
+m
+h
+d
+w
+s
+m
+h
+d
+w
+original
+shuffled
+inter-event times
+Figure 1:
+Top: temporal path entropy as a function of causal temporal scales in datasets Synthetic-
+2, WB-DE, and HC-email (transparent red) and in the temporal networks with shuffled timestamps
+(transparent blue).
+The height of a bar represents temporal path entropy (error bars represent
+the estimation error) and the x-limits of a bar represent the interval τ = [τmin, τmax] on which the
+temporal path entropy was measured. We indicate on x-axis the timescales of one minute (m),
+hour (h), day (d), week (w), and year (y). We observe that the temporal path entropy differs more
+between the original and the shuffled network at causal timescales. Bottom: histogram of causal
+inter-event times.
+In practice, the temporal path entropy can be estimated from the counts of time-respecting
+paths Pk
+τ (Γ) by assuming multinomial distributions with respective probabilities p(v0, . . . , vk−1) and
+p(v0, . . . , vk). The counts of time-respecting paths can be computed e.g. using the methods from
+[Kivel¨a et al., 2018, Petrovi´c and Scholtes, 2021]. The estimation of the entropy can be challenging
+especially for small ranges of timescales, since the temporal network can get temporally disconnected,
+resulting in very few paths of order k being observed. As a result we require an efficient method for
+estimating the entropy that performs well even in such under-sampled regimes. The simplest estima-
+tor of a multinomial distribution, called the plug-in estimator, is based on the maximum likelihood
+estimation, which, however, is known to severely underestimate the entropy in the undersampled
+regime and has various corrections [e.g. Miller, 1955, Grassberger, 2003]). An alternative to the
+plug-in estimator is to follow a Bayesian approach which results in entropy estimators that strongly
+depend on the choice of prior. To counteract this dependency, the NSB estimator [Nemenman et al.,
+2001] directly infers the entropy from the counts by averaging over different priors for the transition
+probabilities, rather than inferring the transition probabilities themselves. Being a Bayesian method,
+the NSB estimator can also be used to quantify the uncertainty of the estimates. Assuming that
+the estimates of H(v0, . . . , vk) and H(v0, . . . , vk−1) have independent errors σk and σk−1, we can
+approximate the total error of the estimate as σ = (σ2
+k + σ2
+k−1)1/2. As the NSB estimator requires
+the size of the alphabet to be known, it is most suitable for cases where the number of nodes is fixed
+and improves further if the set of edges that can occur are known a priory as this further restricts
+the number of potential paths. In cases when the number of nodes in the system is unknown, the
+4
+
+0
+2
+EU-email-A
+100
+101
+102
+103
+104
+105
+106
+0.0
+2.5
+DNC-16
+s
+m
+h
+d
+w
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+time [s]
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+H[ nat ]
+original
+shuffled
+Figure 2: Temporal path entropy as a function of the timescale τ in EU-email-A and DNC-16 and in
+timestamp shuffled networks. Timescale τ is represented with the x-limits of the bar, and temporal
+path entropy is represented as the height of the bar. Error bars indicate the error of the temporal
+path entropy estimates.
+Pitman-Yor Mixture entropy estimator [Archer et al., 2014] could be used instead.
+Finally, we address testing whether an interval τ is a causal timescale of a temporal network Γ.
+To do so, we need to assume the null hypothesis that there are no temporal correlations between
+temporal edges, but the main issue is to obtain a sample of temporal networks under this assump-
+tion. To resolve this issue, we repeatedly randomize the observed temporal network Γ by randomly
+permuting timestamps between its temporal edges [Holme and Saram¨aki, 2012]. These samples of
+temporal networks would preserve both the edge frequencies and timestamp distribution while de-
+stroying the correlations between temporal edges. We can use the samples to determine whether
+temporal path entropy of the observed network has an unexpected value under the null assumption.
+4
+Experiments
+In the following part, we first show the behavior of temporal path entropy in synthetically generated
+temporal networks with known causal timescales (the description of the generation process can
+be found in the Appendix); we then present how it behaves in two real-world networks with the
+information about the ground truth timescales, and two real world networks without the information
+about the ground truth timescales.
+In the top left panel of Fig. 1, we present the temporal path entropy H (y-axis) for various
+timescales (x-axis): the left and right x limit of a bar represents τmin and τmax, and the height of
+the bar represents H. The results are shown both for the synthetic network and its shuffled network.
+In the bottom left panel, we show the histogram of inter-event times on causal paths. We observe
+that the temporal path entropy behaves as expected and decreases in accordance with the planted
+timescale at which the interactions cause one another. Moreover, this pattern disappears when the
+timestamps of edges are shuffled, demonstrating that temporal path entropy captures the interplay
+of temporal and topological patterns.
+We consider here two empirical temporal networks where we have information about the ground
+truth causal structure and two empirical temporal networks where we have no information about
+5
+
+the ground truth causal structure. As a first data set, we consider the bipartite temporal network of
+German Wikibooks co-editing patterns (WB-DE) [Wikimedia Foundation, Peixoto, 2020]. This data
+contains information about edits on the Wikibooks website: for each edit, we know the editor, the
+article that was edited, and the time at which the edit occurred. We preprocess this data to obtain
+a temporal network of editors: if editor v edited an article prior to editor w who edited the same
+article at time t, we assume that a link (v, w, t) occurred in the temporal network of editors. We
+define causal inter-event times based on the articles: we extract the time intervals between successive
+edits of each article. In WB-DE data, we analyze the timescales of the whole temporal network.
+As a second data set, we consider public data set of Hillary Clinton’s emails (HC-email) [Kaggle,
+2022], which contains the sender, the receiver, the timestamp, and the subject of each email. In this
+data set we analyze the timescales of node representing Hillary Clinton. While sender, receiver and
+the timestamp constitute a temporal network, email subjects allow us to obtain causal inter-event
+times: for each incoming email, we extract the time duration until an email with the same subject
+was sent. We use the inter-event times between emails with the same subject and the inter-event
+times of articles for evaluation; the temporal networks contain only the temporal edges and not any
+additional information about the ground truth timescales. We also use two email data sets without a
+ground truth timescales: EU-email-A [Paranjape et al., 2017], which contains email correspondence
+between researchers of an EU institution from four deparments, and DNC-16 [Rossi and Ahmed,
+2015], which contains emails of the US Democratic National Committee. Results on other datasets
+as well as details of all datasets are in the Appendix. Reproducibility package is available at [Petrovi´c
+et al., 2023].
+Results of the WB-DE and HC-email data are in Fig. 1 (middle and right, respectively). When
+we compare the histogram of causal inter-event times with the temporal path entropy at different
+timescales of the temporal network, we see that increased number of causal interactions increases the
+difference in temporal path entropy between the original and the shuffled network. The temporal
+path entropy converges for large timescales because the interval sizes increase, the density of causal
+interactions decreases, and the noise increases. In Fig. 2, although we do not have the ground truth,
+we see that the largest difference between the original and the shuffled datasets are at timescales
+between a minute a day, which is what we would expect from email correspondence.
+We identify four limitations of our approach. First, our base assumption is that the interactions,
+represented by edges, cause one another, and our measure can not separate that case that from
+the case when edges are generated by some common factor. Second, being based on directed paths
+the current method is restricted in the types of causal interactions it considers namely interactions
+where a incoming link into a vertex effects the subsequent links emanating from the vertex. The
+method could potentially be generalized to other types of interactions by considering other patterns
+to alleviate this shortcoming. Third, our method cannot detect timescales at which the incoming
+edges to a node change the overall activity of the node without changing the relative frequencies of
+the outgoing edges. Detecting timescales of such causal influences is thus an open problem. Fourth,
+real data can contain time-varying timescales, e.g. during day or night, which would probably require
+an application of time warping techniques.
+5
+Conclusion
+To summarize, the analysis of temporal networks heavily depends on the analysis of time-respecting
+paths [Holme and Saram¨aki, 2012, Holme, 2015, Pan and Saram¨aki, 2011, Masuda et al., 2013,
+Scholtes et al., 2016, Kivel¨a et al., 2018]. However, in order to model and analyze the time-respecting
+paths, we first need to identify the correct timescale.
+In this work we address this problem by
+6
+
+introducing an information theoretic measure, the temporal path entropy, that is able to can identify
+timescales at which the influences are highly correlated. Using real world data we demonstrated that
+the measure can be applied to temporal networks as a whole as well as to a single node. We showed
+that the temporal path entropy can capture the causal timescales in both synthetic and empirical
+temporal networks. We further support our findings by observing that the differences in the temporal
+path entropy between the original and shuffled networks coincide with increases in the number of
+causal paths. The temporal path entropy allows system-relevant timescales to be inferred from the
+temporal networks themselves which is crucial for the analysis of temporal networks where inherent
+timescales are unavailable and hard to measure.
+Acknowledgments
+The authors would like to thank Christopher Bl¨ocker, Chester Tan, and Franziska Heeg for valu-
+able comments on the manuscript. LP and IS acknowledge support by the Swiss National Science
+Foundation, grant 176938.
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+6
+Datasets
+In this work we considered synthetic and empirical temporal networks.
+To generate synthetic temporal networks Synthetic-1, Synthetic-2 and Synthetic-3 with a ground
+truth timescale ¯τ = [¯τmin, ¯τmax], we start from a static Erd˝os-R´enyi random graph with 50 nodes and
+500 directed edges. We sample a random subset Pcausal of nu.p. = 500 unique paths of length k = 2
+in the static network which correspond to causal influences in the system. We sample with repetition
+np = 5000 paths from Pcausal to generate dataset Synthetic-1, np = 10000 paths to generate dataset
+Synthetic-2 and np = 20000 paths to generate dataset Synthetic-3. To add each path (v0, v1, v2)
+to the temporal network, we sample a random starting time t uniformly from [0, Ttotal − ¯τmax] and
+create a temporal edge (v0, v1, t); we then sample temporal distance δ between edges on the path
+(inter-event time) uniformly from ¯τ and create the temporal edge (v1, v2, t + δ). We choose ¯τ with
+¯τmin = 100 and ¯τmax = 200. To add some noise to the system, we uniformly sample 20000 edges
+from the static graph, and sample their timestamps uniformly from [0, Ttotal]. The temporal network
+Synthetic-4 contains two time-scales relevant for the dynamics. To do so, we generated two different
+temporal networks based on two random graphs of 50 nodes (with the same node names) and 500
+edges and based on the different timescales τ 1 = [50, 100] and τ 2 = [150, 200]. We used the same
+procedure as above with parameters nu.p. = 500; np = 5000; Ttotal = 105; nr.e. = 10000. We merged
+the two temporal networks into one; the details of the resulting network are in Table 1. The dataset
+Synthetic-5 contains paths of length three. Again, there are 50 nodes and 500 edges in the static
+Erd˝os R´enyi graph.
+We sample nu.p. = 20 unique paths, we sample np = 20000 of them, and
+spread them across Ttotal = 105 using the same procedure and timescale τ = [100, 200]. We add
+nr.e = 10000 random edges to the network as noise.
+We also use empirical dataset where can get access to the ground truth causal path structure. We
+consider the bipartite temporal network of Wikibooks co-edits in Arabic (WB-AR), French (WB-
+FR) and German (WB-DE) [Wikimedia Foundation, Peixoto, 2020]. This data contains information
+about edits on the Wikibooks website: for each edit, we know the editor, the article that was edited,
+and the time at which the edit occurred. We preprocess this data to obtain a temporal network
+of editors: if editor v edited an article prior to editor w who edited the same article at time t, we
+assume that a link (v, w, t) occurred in the temporal network of editors. We define causal inter-event
+times based on the articles: we extract the time intervals between successive edits of each article.
+In these data, we analyze the timescales of the whole temporal network. Another dataset where we
+10
+
+dataset
+|V |
+|E|
+|E|
+Ttotal [s]
+Ants-1-1
+89
+947
+1911
+1.44e+03
+Ants-1-2
+72
+862
+1820
+1.75e+03
+Ants-2-1
+71
+636
+975
+1.44e+03
+Ants-2-2
+69
+769
+1917
+1.8e+03
+Ants-3-1
+11
+37
+78
+1.13e+03
+Ants-3-2
+6
+21
+104
+1.42e+03
+DNC-16
+1891
+5598
+39264
+8.49e+07
+EU-email-1
+309
+3031
+61046
+6.94e+07
+EU-email-2
+162
+1772
+46772
+6.94e+07
+EU-email-3
+89
+1506
+12216
+6.93e+07
+EU-email-4
+142
+1375
+48141
+6.94e+07
+EU-email-A
+986
+24929
+332334
+6.95e+07
+Gallery
+10972
+89034
+831824
+6.95e+06
+HC-email
+326
+385
+8313
+1.19e+08
+Hospital
+75
+2278
+64848
+3.48e+05
+Hypertext
+113
+4392
+41636
+2.12e+05
+OSS
+5789
+6888
+12583
+3.54e+08
+Primary
+242
+16634
+251546
+1.17e+05
+School-13
+327
+11636
+377016
+3.64e+05
+Synthetic-1
+50
+500
+30000
+1e+05
+Synthetic-2
+50
+500
+40000
+1e+05
+Synthetic-3
+50
+500
+60000
+1e+05
+Synthetic-4
+50
+898
+40000
+1e+05
+Synthetic-5
+50
+500
+50000
+1e+05
+WB-AR
+1124
+3334
+27166
+3.89e+08
+WB-DE
+10999
+54700
+464089
+4.87e+08
+WB-FR
+9735
+53606
+362094
+4.88e+08
+Work-13
+92
+1510
+19654
+9.88e+05
+Table 1: The sizes of the sets of nodes V , unique edges E, and temporal edges E of temporal
+networks that we analyzed in the experiments. Datasets synth-2, HC email and WB DE are in the
+main paper. The other datasets are shown in the Appendix.
+can get access to the ground truth causal structure is the public data set of Hillary Clinton’s emails
+(HC-email) [Kaggle, 2022], which contains the sender, the receiver, the timestamp, and the subject
+of each email. In this data set we analyze the timescales of node representing Hillary Clinton. While
+sender, receiver and the timestamp form a temporal network, email subjects allow us to obtain
+causal inter-event times: for each incoming email, we extract the time duration until an email with
+the same subject was sent. We use the inter-event times between emails with the same subject and
+the inter-event times of articles for evaluation; the temporal networks contain only the temporal
+edges and not any additional information about the ground truth timescales. The details of each
+data-set are in Table 1.
+Finally, we also use empirical temporal networks where we do not know the ground truth causal
+path structure. Datasets Ants-1-1, Ants-1-2, Ants-2-1, Ants-2-2, Ants-3-1, and Ants-3-3 [Blonder
+and Dornhaus, 2011] contain antenna contacts in ant colonies. Dataset DNC-16 [Rossi and Ahmed,
+2015] contains emails of the US Democratic National Committee leaked in 2016.
+Datasets EU-
+11
+
+email-1, EU-email-2, EU-email-3, EU-email-4, and EU-email-A [Paranjape et al., 2017] contain
+email correspondence between researchers of an EU institution from first, second, third, fourth and
+all departments, respectively. Datasets Gallery [Isella et al., 2011], Hospital [Vanhems et al., 2013],
+Hypertext [Isella et al., 2011], Primary [Gemmetto et al., 2014, Stehl´e et al., 2011], Work-13 [G´enois
+et al., 2015] and School-13 [Mastrandrea et al., 2015] contain human face-to-face interactions in
+different settings measured by the SocioPatterns collaborations. Dataset OSS [Zanetti et al., 2013]
+contains ASSIGN relationships between members of the Open Source Software community Apache.
+7
+Results: Synthetic Data
+We present results for datasets Synthetic-1, Synthetic-3, Synthetic-4, Synthetic-5.
+1.5
+2.0
+H[ nat ]
+Synthetic-1
+50
+100
+150
+200
+250
+time [s]
+0.0
+0.2
+0.4
+counts [103]
+original
+shuffled
+inter-event times
+Figure 3: Top: temporal path entropy as a function of the timescale τ in temporal network Synthetic-
+1 and in Synthetic-1 with shuffled timestamps. Timescale τ is represented with the x-limits of the
+bar, and temporal path entropy is represented as the height of the bar. Error bars indicate the error
+of the temporal path entropy estimates. Bottom: histogram of inter-event times of synthetic causal
+interactions.
+8
+Results: Empirical Data with Ground Truth
+In this section we show results on other Wikibooks datasets that we used to test the method. In
+Fig. 7, we test temporal path entropy on the WB-AR dataset, and in Fig. 8, we test it on the WB-FR
+dataset. Similar to the WB-DE in the main paper, the bottom panel shows the yellow histogram of
+inter-event times of edits per article for all articles.
+9
+Empirical data without the ground truth
+In this section, we show multiple datasets in which we do not have access to the ground truth
+temporal scales. Although the lack of ground truth in these datasets makes objective evaluation
+12
+
+1.5
+2.0
+H[ nat ]
+Synthetic-3
+50
+100
+150
+200
+250
+time [s]
+0
+1
+2
+counts [103]
+original
+shuffled
+inter-event times
+Figure 4: Equivalent of Fig. 3, for Synthetic-3.
+2
+3
+H[ nat ]
+Synthetic-4
+50
+100
+150
+200
+250
+time [s]
+0.0
+0.5
+1.0
+counts [103]
+original
+shuffled
+inter-event times
+Figure 5: Equivalent of Fig. 3, for Synthetic-4.
+of the method difficult, the results across datasets are consistent and in accordance with what one
+would expect: e.g. in the email datasets, temporal path entropy is different between the original
+and the shuffled network for timescales between one minute and a few days, which corresponds to
+what we would expect to be the interval in which emails are responded to.
+13
+
+1.0
+1.2
+H[ nat ]
+k = 2
+Synthetic-5
+0.75
+1.00
+1.25
+H[ nat ]
+k = 3
+50
+75
+100
+125
+150
+175
+200
+225
+250
+time [s]
+0.0
+2.5
+counts [103]
+original
+shuffled
+inter-event times
+Figure 6: Temporal path entropy as a function of the timescale τ in temporal network Synthetic-5
+and in Synthetic-5 with shuffled timestamps for orders k = 2 (top) and k = 3 (middle). Timescale τ
+is represented with the x-limits of the bar, and temporal path entropy is represented as the height of
+the bar. Error bars indicate the error of the temporal path entropy estimates. Bottom: histogram
+of inter-event times of synthetic causal interactions.
+0.0
+0.5
+1.0
+H[ nat ]
+WB-AR
+100
+102
+104
+106
+time [s]
+0
+2
+counts [103]
+s
+m
+h
+d
+w
+original
+shuffled
+inter-event times
+Figure 7:
+Top: temporal path entropy as a function of the timescale τ in WB-AR temporal network
+and of WB-AR temporal network with shuffled timestamps. Timescale τ is represented with the
+x-limits of the bar, and temporal path entropy is represented as the height of the bar. Error bars
+indicate the error of the temporal path entropy estimates. Bottom: histogram of inter-event times
+for all articles of edits of the same article.
+14
+
+0.0
+0.5
+1.0
+H[ nat ]
+WB-FR
+100
+102
+104
+106
+time [s]
+0
+20
+counts [103]
+s
+m
+h
+d
+w
+original
+shuffled
+inter-event times
+Figure 8:
+Equivalent of Fig. 7 for WB-FR.
+15
+
+100
+101
+102
+103
+time [s]
+0
+1
+2
+3
+4
+5
+H[ nat ]
+Ants-1-1
+original
+shuffled
+s
+m
+100
+101
+102
+103
+time [s]
+0
+1
+2
+3
+4
+H[ nat ]
+Ants-1-2
+original
+shuffled
+s
+m
+100
+101
+102
+103
+time [s]
+0
+1
+2
+3
+4
+H[ nat ]
+Ants-2-1
+original
+shuffled
+s
+m
+100
+101
+102
+103
+time [s]
+0
+1
+2
+3
+H[ nat ]
+Ants-2-2
+original
+shuffled
+s
+m
+100
+101
+102
+103
+time [s]
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+H[ nat ]
+Ants-3-1
+original
+shuffled
+s
+m
+100
+101
+102
+103
+time [s]
+0.0
+0.5
+1.0
+1.5
+2.0
+H[ nat ]
+Ants-3-2
+original
+shuffled
+s
+m
+Figure 9: Temporal path entropy as a function of the timescale τ in temporal networks of antenna
+contacts in ant collonies. For each temporal network, we show the temporal path entropy of the
+original and of a shuffled network. Timescale τ is represented with the x-limits of the bar, and
+temporal path entropy is represented as the height of the bar. Error bars indicate the error of the
+temporal path entropy estimates.
+16
+
+100
+101
+102
+103
+104
+105
+106
+time [s]
+0
+1
+2
+3
+4
+H[ nat ]
+DNC-16
+original
+shuffled
+s
+m
+h
+d
+w
+100
+101
+102
+103
+104
+105
+106
+time [s]
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+H[ nat ]
+EU-email-A
+original
+shuffled
+s
+m
+h
+d
+w
+100
+101
+102
+103
+104
+105
+106
+time [s]
+0
+1
+2
+3
+H[ nat ]
+EU-email-1
+original
+shuffled
+s
+m
+h
+d
+w
+100
+101
+102
+103
+104
+105
+106
+time [s]
+0
+1
+2
+3
+4
+5
+H[ nat ]
+EU-email-2
+original
+shuffled
+s
+m
+h
+d
+w
+100
+101
+102
+103
+104
+105
+106
+time [s]
+0
+1
+2
+3
+4
+5
+H[ nat ]
+EU-email-3
+original
+shuffled
+s
+m
+h
+d
+w
+100
+101
+102
+103
+104
+105
+106
+time [s]
+0
+1
+2
+3
+4
+5
+6
+H[ nat ]
+EU-email-4
+original
+shuffled
+s
+m
+h
+d
+w
+Figure 10: Temporal path entropy as a function of the timescale τ in temporal networks of email
+correspondence. For each temporal network, we show the temporal path entropy of the original and
+of a shuffled network. Timescale τ is represented with the x-limits of the bar, and temporal path
+entropy is represented as the height of the bar. Error bars indicate the error of the temporal path
+entropy estimates.
+17
+
+102
+103
+104
+time [s]
+0
+2
+4
+6
+H[ nat ]
+Gallery
+original
+shuffled
+m
+h
+102
+103
+104
+105
+time [s]
+0
+1
+2
+3
+4
+5
+6
+H[ nat ]
+School-13
+original
+shuffled
+m
+h
+d
+102
+103
+104
+105
+time [s]
+0
+1
+2
+3
+4
+5
+6
+H[ nat ]
+Hospital
+original
+shuffled
+m
+h
+d
+102
+103
+104
+105
+time [s]
+0
+1
+2
+3
+4
+5
+6
+H[ nat ]
+Hypertext
+original
+shuffled
+m
+h
+d
+102
+103
+104
+105
+time [s]
+0
+2
+4
+6
+H[ nat ]
+Primary
+original
+shuffled
+m
+h
+d
+102
+103
+104
+105
+time [s]
+0
+1
+2
+3
+4
+5
+H[ nat ]
+Work-13
+original
+shuffled
+m
+h
+d
+w
+Figure 11: Temporal path entropy as a function of the timescale τ in temporal networks of human
+face-to-face interactions measured by the SocioPatterns collaboration. For each temporal network,
+we show the temporal path entropy of the original and of a shuffled network.
+Timescale τ is
+represented with the x-limits of the bar, and temporal path entropy is represented as the height
+of the bar. Error bars indicate the error of the temporal path entropy estimates.
+18
+
+100
+101
+102
+103
+104
+105
+106
+time [s]
+0
+1
+2
+3
+4
+H[ nat ]
+OSS
+original
+shuffled
+s
+m
+h
+d
+w
+Figure 12: Temporal path entropy as a function of the timescale τ in temporal networks ASSIGN
+relationships between members of the Open Source Software community Apache.
+We show the
+temporal path entropy of the original and of a shuffled network. Timescale τ is represented with the
+x-limits of the bar, and temporal path entropy is represented as the height of the bar. Error bars
+indicate the error of the temporal path entropy estimates.
+19
+
+10
+Conditional entropy: The chain rule
+For discrete random variables X and Y , the definition of the entropy (in nats) is
+H(X) = −
+�
+x
+p(X = x) ln p(X = x)
+and the definition of conditional entropy (in nats) H(Y |X) is:
+H(Y |X) = −
+�
+x,y
+p(X = x, Y = y) ln p(X = x, Y = y)
+p(X = x)
+In the following, we use the above definitions to derive the chain rule of conditional entropy:
+H(Y |X) = −
+�
+x,y
+p(X = x, Y = y) (ln p(X = x, Y = y) − ln p(X = x)) =
+= −
+�
+x,y
+p(X = x, Y = y) ln p(X = x, Y = y) −
+�
+−
+�
+x,y
+p(X = x, Y = y) ln(p(X = x)))
+�
+=
+= H(X, Y ) −
+�
+−
+�
+x,y
+p(Y = y|X = x)p(X = x) ln(p(X = x)))
+�
+=
+= H(X, Y ) −
+�
+�−
+�
+x
+p(X = x) ln(p(X = x)))
+
+:1
+��
+y
+p(Y = y|X = x)
+�
+�
+� =
+= H(X, Y ) − H(X).
+(5)
+20
+

HtE1T4oBgHgl3EQfrgVp/content/tmp_files/2301.03355v1.pdf.txt

@@ -0,0 +1,1489 @@
+Charge transfer mediated giant photo-amplification in air-stable α-CsPbI3 
+nanocrystals decorated 2D-WS2 photo-FET with asymmetric contacts 
+Shreyasi Das1, Arup Ghorai1,2, Sourabh Pal3, Somnath Mahato1, Soumen Das4, Samit K. Ray5 * 
+1School of Nano Science and Technology, IIT Kharagpur, Kharagpur, West Bengal, India, 721302 
+2Department of Materials Science and Engineering, Pohang University of Science and Technology, 
+Pohang 790-784, Korea 
+3Advanced Technology Development Centre, IIT Kharagpur, Kharagpur, West Bengal, India, 721302 
+4School of Medical Science and Technology, IIT Kharagpur, Kharagpur, West Bengal, India, 721302 
+5Department of Physics, IIT Kharagpur, Kharagpur, West Bengal, India, 721302 
+Email : physkr@phy.iitkgp.ac.in 
+ 
+Abstract 
+Hybrid heterostructure based phototransistors are attractive owing to their high gain induced 
+by photogating effect. However, the absence of an in-plane built-in electric field in the single 
+channel layer transistor results in a relatively higher dark current and require a large operating 
+gate voltage of the device. Here, we report novel air-stable cesium lead iodide/tungsten di-
+sulfide (CsPbI3/WS2) mixed dimensional heterostructure based photo-field-effect-transistors 
+(photo-FETs) with asymmetric metal electrodes (Cr/WS2/Au), exhibiting extremely low dark 
+current (~10-12 A) with a responsivity of ~ 102 A/W at zero gate bias. The Schottky barrier 
+(WS2/Au interface) induced rectification characteristics in the channel accompanied by the 
+excellent photogating effect from solution-processed α-phase CsPbI3 NCs sensitizers, resulting 
+in gate-tunable broadband photodetection with a very high responsivity (~104 A/W) and 
+excellent sensitivity (~106). Most interestingly, the device shows superior performance even 
+under high humidity (50-65%) conditions owing to the formation of cubic α-phase CsPbI3 
+nanocrystals with a relatively smaller lattice constant (a = 6.2315 Å) and filling of surface 
+vacancies (Pb2+ centres) with the sulfur atoms from WS2 layer, thus protecting it from 
+environmental degradation. These results emphasise a novel strategy for developing mixed 
+dimensional hybrid heterostructure based phototransistors for futuristic integrated nano-
+optoelectronic systems.  
+Keywords: Two dimensional TMDs, Inorganic perovskites, Sensitizers, Asymmetric 
+electrodes, Mixed dimensional phototransistors 
+ 
+ 
+
+Introduction 
+Fabrication of high performance phototransistors demands superior channel material, which 
+should be of high carrier mobility for high gain bandwidth product, a direct bandgap for 
+efficient optical absorption, a thinner layer for full depletion leading to ultralow dark current 
+and very low trap state density for low subthreshold swings. Layered semiconducting two 
+dimensional (2D) TMDs fulfil most of these requirements [1–4] making them a potential 
+candidate for photo-field-effect transistor (photo-FET). However, some trade-offs are found in 
+single semiconductor channel based phototransistors, due to the simultaneous occurrence of 
+both absorption and amplification processes within the same layer, compromising the device 
+performance [5–7]. To overcome this shortcoming as well as for fabrication simplicity, 
+attention has been paid on few-layer TMDs over monolayer with a wider spectral 
+photoresponse and relatively higher absorbance [8–10], which in turn increases the dark current 
+values requiring a higher gate voltage to operate the device in full depletion mode [11]. This 
+issue can be addressed via incorporating a Schottky barrier by judiciously selecting the metal 
+contacts and utilizing the developed depletion region to facilitate an unidirectional current 
+transport, which leads to the significant lowering of dark current in few layered TMD based 
+photo-FETs at zero gate bias [12–14]. The built-in electric field of the Schottky interface 
+further helps in the separation of photogenerated charge carriers across the channel  layer [15] 
+leading to zero gate bias driven enhanced photosensitivity, although the responsivity of these 
+devices are limited owing to the lower absorption coefficient of 2D TMD material. For the 
+further improvement of photoresponsivity, recent works are focused on sensitizing the thin 
+channel material with semiconductor nanocrystals (NCs), referred to as sensitizers, having 
+excellent absorption characteristics and opposite doping polarity to create a vertical junction 
+for subsequent charge separation [16–19].  
+
+Recently, all-inorganic cesium lead halide (CsPbX3: X = I, Br, and Cl) perovskite NCs have 
+drawn  tremendous interests in the field of optoelectronics [20], featuring superior emissive 
+(approaching photoluminescence quantum yield ~ 100%) [21] characteristics, extremely high 
+absorption coefficient [22], large carrier diffusion lengths (>3 mm), fast radiative 
+recombination rates and most importantly their improved stability [23] over organic-inorganic 
+hybrid perovskites [24,25]. Especially colloidal synthesised zero-dimensional (0D) α-CsPbI3 
+NCs exhibit extended photoabsorption band covering the whole visible spectrum with high 
+absorption coefficient (~ 3 × 104 cm-1 at 680 nm) [26] and  better photostability, making them 
+attractive as a photoactive material for high performance optoelectronic devices under ambient 
+condition.   Weerd et al. have recently reported colloidal CsPbI3 NCs with high quantum yield 
+of ~ 98% prepared by hot-injection method which reveals highly efficient carrier multiplication 
+as well as longer build-up of free carrier concentration [27]. Among four existing phases (α-
+cubic, β-tetragonal, γ-orthorhombic, and δ-orthorhombic) of CsPbI3 NCs, cubic α-phase has 
+high stability due to its low surface-to-volume ratio (lattice constant a = 6.23 Å) without any 
+octahedral inclination or lattice distortion in the [PbI6]4− octahedra as well as in the unit 
+cell [28–30]. These extraordinary properties of α-phase CsPbI3 NCs provoke to utilize them as 
+an effective sensitizer in 2D channel based hybrid phototransistor devices. However, owing to 
+the vulnerability towards moisture for halide perovskite NCs, recently, the surface passivation 
+of CsPbI3 via coordination of Pb2+ centres with sulphur donors has been explored to protect 
+them from environmental degradation [31] to improve the device stability under ambient 
+conditions. 
+In this work, we present a proof-of-concept for 0D/2D CsPbI3/WS2 based mixed dimensional 
+van der Waals heterostructure (MvWH) photo-FETs utilizing the superior photoabsorption 
+attributes of α-phase CsPbI3 NCs as sensitizers, along with a sub 5-nm thick 2D WS2 channel 
+which acts as an expressway for carrier transport. In our device configuration, the developed 
+
+built-in electrical field at the heterostructure interface facilitates efficient transfer of 
+photogenerated carriers from CsPbI3 NCs into the WS2 layer resulting in an excellent 
+broadband visible photoabsorption in the hybrid system. Moreover, asymmetric metal contacts 
+(Au-Cr) as source and drain electrodes are purposefully chosen to utilize the large built-in 
+potential along the WS2/Au Schottky junction for a diode-like unidirectional current flow and 
+effective separation of photogenerated electron-hole pairs, leading to an excellent rectifying 
+ratio of ~ 104 with a very low dark current of the order of ~ pA under a low source-to-drain 
+bias (VDS) without any applied back gate voltage.  Fabricated hybrid phototransistor devices 
+exhibit a very high photoresponsivity of ~ 104 A/W at ~ 40 V gate bias upon visible light 
+illumination (~ 0.1 µW) due to the photogating effect and a broad spectral bandwidth across 
+the entire visible range. In addition, the coordination of sulphur atoms of WS2 layer with Pb2+ 
+centres of CsPbI3 NCs reveals an excellent device stability under 50-65% humidity condition. 
+Our results not only open up new avenues for studying fundamental carrier transport and 
+relaxation pathways in hybrid van der Waals heterojunctions, but also pave the way for 
+constructing high-performance optoelectronic devices using mixed-dimensional 0D/2D hybrid 
+building blocks, rather than using purely 2D layered materials. 
+Experimental section: 
+Synthesis of α-phase CsPbI3 NCs:  
+This is a two-step reaction, where, in the first step Caesium oleate (Cs-OA) was prepared 
+followed by the synthesis of α-phase CsPbI3 NCs in the second step.  
+i) Firstly, caesium carbonate (Cs2CO3) of 814 mg and 40 ml Olive oil were added in a round 
+bottom two-neck flask, which was heated at 120⁰C for 1 hr under vacuum condition. 
+Followed by the rise in temperature to 150⁰C under N2 atmosphere for 10-15 mins, the 
+desired transparent solution of Cs-OA was stored for further use to synthesize CsPbI3.    
+ii) Next, 870 mg of PbI2 and 5 ml of olive oil (instead of 1-octadecene (ODE)) were mixed 
+which was heated to 120⁰C under vacuum for 1hr. Thereafter, we swiftly injected 1 ml of 
+
+Oleyl amine (OLAm) in the reaction mixture under N2 atmosphere to get a transparent 
+solution. Finally, preheated (100⁰C) Cs-OA was injected to the reaction mixture and cooled 
+immediately in an ice bath to quench the reaction to obtain desired α-phase CsPbI3 NCs.  
+Finally, as-synthesized CsPbI3 was purified through centrifugation using excess hexane. The 
+centrifugation process (20000 rpm) was repeated for several times to remove excess 
+OLAm/olive oil from the product. Finally, the sedimentation was collected and re-dispersed in 
+hexane. The collected dispersion was stored in a sealed vial for further characterisations and 
+device fabrications. 
+CsPbI3/WS2 MvWH photo-FET device fabrication:  
+For the fabrication of the CsPbI3/WS2 MvWH photo-FET device, WS2 flakes were 
+mechanically exfoliated from the bulk WS2 crystal (2D semiconductor Inc. Scottsdale, AZ, 
+USA) using a Scotch tape (3M Inc. USA) on polydimethylsiloxane (PDMS) gel film (Gel-Pak 
+Inc. Hayward, CA, USA) and the interested few layer flakes were identified under optical 
+microscope, followed by the layer confirmation via Raman characteristics and AFM height 
+profile. Note that, mechanically exfoliated flakes were chosen for their high quality and clean 
+interface promising greater mobility and high performance device fabrication. Further, for the 
+Au contact with WS2, electrodes were patterned in advance on a pre-patterned SiO2/Si (285 nm 
+oxide thickness) substrate using e- beam lithography technique and then Cr (5 nm)/Au (30 nm) 
+were deposited via e-beam evaporation followed by lift-off with acetone. Then, the selected 
+few layer flakes were deterministically transferred on the targeted Au electrodes from PDMS 
+gel film following the dry transfer technique to make sure residue free clean transfer. After the 
+successful transfer of the flakes on Au electrodes, again electrodes were patterned using second 
+step e-beam lithography followed by metal deposition Cr (5 nm)/Au (30 nm) for Cr contacts 
+on WS2. To remove the resist residue and improve the contact conductance, the fabricated 
+devices were annealed at 150°C for 2 hrs in a high vacuum of ~ 10−3 Torr. Finally, the 
+synthesized diluted solution (0.1 mg mL−1) of perovskite NCs was uniformly spin-coated 
+several times (varying from one to four) onto WS2 layer with a speed of 2000 rpm.  
+
+Characterisations and measurements:  
+X-ray diffraction (XRD, Philips MRD X-ray diffractometer) patterns were recorded using 
+characteristic Cu-Kα (λ = 1.5418 Å) radiation with 2.0° grazing incidence angle. For the 
+Transmission electron microscopy (TEM) sample preparation, CsPbI3 NCs solution was 
+dissolved in Hexane and then placed into a TEM grid and dried it for few minutes and did the 
+measurement. The TEM images were carried out using TECNAI G2 TF20-ST and JEM-2100F 
+Field Emission Electron Microscope operating at 200 kV equipped with Gaytan’s latest CMOS 
+camera. All the images were proceeding by Digital Micrograph Software for the estimation of 
+d-Spacing & Indexing. UV–vis–NIR absorption spectra of as synthesised CsPbI3 samples were 
+recorded using a fiber probe-based UV–vis–NIR spectrophotometer (Model: U-2910 
+Spectrophotometer, HITACHI) and a broadband light source. Raman and PL spectra were 
+recorded using a semiconductor laser of excitation wavelength 532 nm, equipped with a CCD 
+detector, an optical microscope of 100x objective lens and a spectrometer (WITec alpha-300R). 
+The photogenerated carrier lifetime was measured by exciting the material with a pulsed diode 
+laser of wavelength 372 nm and detecting the signal using Edinburgh LifeSpec-II fluorescence 
+lifetime spectrometer fitted with a PMT detector. Room-temperature current−voltage 
+characteristics were recorded using a Keithly semiconductor parameter analyzer (4200 SCS) 
+in the presence of an Argon laser (514 nm) and a broadband solar simulator (AM 1.5, 100 
+mW/cm2) as a visible light source. 
+Results and discussion 
+
+ 
+FIG. 1. (a) Rietveld refinements (α-phase fitting) of the XRD pattern of a film of cubic CsPbI3 
+NCs. (b) 1×1 3D VESTA visualization image of α-CsPbI3 cubic crystal structure. (c) Typical 
+HRTEM image of CsPbI3 NCs with an energy 200 keV revealing cubic morphology. (d) FFT 
+patterns from a region marked by the red dotted square on the micrograph (c). (e) A magnified 
+view of the corresponding HRTEM image in the selected yellow square region on micrograph 
+(c). (f) SAED pattern of α-CsPbI3 NCs showing well defined diffraction spots indexed as (200), 
+(220) and (020) planes viewed along [004] zone axis. 
+ 
+To study the crystal structure of as-synthesized CsPbI3 NCs, we have recorded X-ray 
+diffraction pattern, followed by their fitting with Rietveld refinement full proof software, which 
+are presented in Fig. 1(a). An excellent agreement with fitted results indicates the growth of 
+single-phase (α-phase) cubic CsPbI3. The crystal structure of CsPbI3 NCs visualized using 
+VESTA 3D software through Rietveld fitting is shown in Fig. 1(b). The VESTA 3D (1x1) 
+structure shows the absence of any octahedral inclination in the perovskite NCs, which is 
+known to be beneficial for achieving higher stability under laboratory ambient (45-50% 
+humidity). Typical high resolution transmission electron microscopy (HRTEM) image reveals 
+almost cubic shape of the synthesised NCs (15.05 nm × 18.04 nm), as shown in Fig. 1(c). 
+Corresponding first Fourier transform (FFT) pattern presented in Fig. 1(d) of the red squared 
+
+(a)
+C
+(200)
+&
+Observed
+(100)
+米0
+Calculated
+15.05nm
+Intensity (arb. units)
+(210)
+Difference
+d= 0.62 nm
+Braggposition
+11001
+d= 0.62 nm
+[100]
+10nm
+10
+20
+20 (degree)
+30
+40
+50
+(b)
+(d)
+ZA[002]
+a-CsPbl3
+f)
+ZA [004]
+a-CsPbl
+(220)
+(210)
+(200)
+(200)
+(200)
+(210)
+(220)
+(220) ( ()
+2 nm
+2 1/nm
+(020)
+Cs
+Pbportion of the Fig. 1(c), shows pure cubic α-phase pattern along the zone axis [002]. Whereas, 
+the high-resolution fringe pattern from the yellow squared region of Fig. 1(c) shows a d-spacing 
+of 0.62 nm, which is in well matched with the cubic α-phase of CsPbI3 [30], as shown in Fig. 
+1(e). The result indicates (100) directional growth of cubic phase CsPbI3 NCs, which is in well 
+agreement with our previously reported results [30]. Corresponding selected area electron 
+diffraction (SAED) patterns shown in the Fig. 1(f), with indexed (200), (220) and (020) planes 
+along the zone axis [004], also corroborate the pure cubic structure of synthesized CsPbI3.      
+Figure 2(a) presents the optical absorption and emission properties of the as-synthesised α-
+phase CsPbI3 NCs in the visible wavelength range with an absorption maxima at ~ 680 nm and 
+the corresponding bandgap value is ~ 1.814 eV [30], extracted from the Tauc plot shown in 
+Fig. S1 within the Supplimental Material. Further, the photoluminescence (PL) maxima at ~ 
+687 nm confirms the formation of excitons (Xα) across the direct bandgap (∼1.80 eV) of α-
+phase CsPbI3 NCs represented via blue curve in Fig. 2(a). The Gaussian line shape of the PL 
+spectrum and the absence of any other PL peaks clearly dictate the synthesis of pure α-phase 
+CsPbI3 without presence of any mixed phase. To examine the charge transfer mechanism at the 
+CsPbI3 NCs/WS2 interface, Raman spectroscopy and micro-PL (µ-PL) measurements have 
+been carried out by spin-coating of a dilute solution of CsPbI3 NCs uniformly on the exfoliated 
+WS2 surface. For the room temperature µ-Raman-PL measurements, samples have been 
+excited with a CW laser having a wavelength of 532 nm with the laser power being kept at a 
+very low value to avoid any local heating induced sample degradation. Figure 2(b) represents 
+comparative Raman spectra of WS2 and mixed dimensional van der Waals heterostructure 
+(MvWH) samples, showing intense in-plane 2LA+E12g Raman modes at ∼ 351 cm–1 and out-
+of-plane vibrational A1g peaks at ∼ 420 cm–1  [32]. The A1g vibrational Raman mode, which 
+preserves the symmetry of the lattice, is clearly red-shifted by ∼ 4.7 cm-1 in case of CsPbI3 
+decorated WS2 layer [inset of Fig. 2(b)], revealing the interfacial charge transfer phenomena. 
+
+The external electron doping in 2D WS2 leads to the filling-up of antibonding states of the 
+conduction band, mostly made up of d z2 orbitals of transition metal atoms [33]. This makes the 
+bonds weaker and the A1g peak of pristine WS2 is shifted towards a lower wavenumber on 
+significant electron doping from CsPbI3 NCs [34]. 
+ 
+FIG. 2. (a) Absorption (Green) and photoluminescence (Blue) spectra of as-synthesised cubic 
+phase CsPbI3 NCs. (b) Comparative Raman spectra of WS2 flakes before and after CsPbI3 NCs 
+decoration showing characteristic E2g and A1g peaks of layered WS2. Inset shows the magnified 
+image of the out-of-plane A1g mode revealing a clear peak shift to lower wavenumbers due to 
+electron doping in WS2 flakes from CsPbI3 NCs. (c) Deconvoluted PL spectra of (i) a bare ML 
+WS2 flake and (ii-v) the heterostructure samples with varying number of spin coated layers of 
+CsPbI3 NCs on the WS2 flake. The spectra (ii), (iii), (iv) and (v) represent the PL emission from 
+the first, second, third and fourth spin coated layers of CsPbI3 NCs, respectively. The green 
+(red) peak represents the A excitonic (A- trionic) emission from ML WS2 flakes and the blue 
+peak represents the emission from band to band transition of cubic α-phase CsPbI3 NCs. (d) 
+Energy band diagram of CsPbI3/WS2 hybrid heterostructures showing effective electron doping 
+in WS2 from CsPbI3 sensitizers and hole trapping in the NCs giving rise to a strong photogating 
+effect.  (e) Schematic representation of the charge transfer mechanism in 0D/2D CsPbI3/WS2 
+hybrid heterostructures giving rise to trion formation in ML WS2. (f) Normalised time resolved 
+PL decay curves of CsPbI3 NCs (Blue curve) and CsPbI3/WS2 hybrids (Red curve), measured 
+using an excitation wavelength of 372 nm. 
+ 
+On the other hand, the monolayer (ML) WS2 PL emission characteristics [Fig. 2c(i)] consist of 
+a strong A-excitonic emission at ∼ 1.995 eV, corresponding to the direct band-to-band 
+550
+600
+650
+700
+750
+ 
+ 
+ 
+ 
+PL intensity (arb. units)
+Wavelength (nm)
+ 
+ 
+ 
+ 
+(v)
+(iv)
+(iii)
+(ii)
+ 
+ 
+(i)
+WS2
+A0
+160
+180
+200
+220
+240
+tav = 33.7 ns
+PL (arb. units)
+Time (ns)
+CsPbI3
+CsPbI3/WS2
+tav = 40.2 ns
+300
+400
+500
+410 420 430
+ 
+ 
+ 
+4.7 cm-1
+400
+410
+420
+430
+440
+ 
+ 
+ 
+4.7 cm-1
+CsPbI3/WS2
+ 
+ 
+Intensity (arb. units)
+Raman shift (cm-1)
+WS2
+Xα
+A-
+CsPbI3/WS2
+500
+600
+700
+800
+Wavelength (nm)
+Nor. Abs./PL (arb. units)
+Absorption
+Photoluminescence
+WS2
+CsPbI3
+hν
+Electron 
+transfer
+Hole 
+trapping
+Charge
+transfer
+WS2
+Exciton
+Trion
+CsPbI3
+hν
+Electrons
+Holes
+(d)
+(c)
+(a)
+(f)
+(b)
+(e)
+Increasing CsPbI3 coating layer
+
+transitions at the K (and/or K’) point of the Brillouin zone, and a weaker trionic A– emission at 
+∼ 1.965 eV with a binding energy of ∼ 30 meV, which are in good agreement with the 
+previously reported results  [34]. It is to be noted that, ML WS2 is purposefully chosen for the 
+charge transfer study via PL measurements due to its extraordinary luminescence property at 
+room temperature owing to the direct bandgap transition. The existence of trions in the room 
+temperature emission spectrum indicates the unintentional doping in the un-passivated WS2 
+flake from the substrate as well as the surrounding environment [35,36]. A systematic PL study 
+of the hybrid structure with increasing layer numbers of CsPbI3 NCs spin-coated over ML WS2 
+shows a pronounced excitonic-PL quenching in MvWH as compared to both the pristine 
+materials (Fig. S2 within the Supplimental Material). Further, the CsPbI3/WS2 MvWHs show 
+relatively broad PL spectra with combined contributions from CsPbI3 NCs as well as ML WS2 
+and the spectral shape changes with increasing CsPbI3 spin-coated layer numbers [Figs. 2c(ii-
+v)]. To explore the effect of CsPbI3 coating over WS2, we have fitted each spectrum with three 
+Gaussian peaks containing the characteristics excitonic features of both the materials and 
+analysed their intensity variation with increasing concentration of CsPbI3 treated on WS2 
+flakes, as shown in Fig. 2(c). It is noticed that the distinctive trion peak (A–) of WS2 becomes 
+prominent with increasing density (coating number) of CsPbI3 NCs and finally excitonic to 
+trionic (integrated intensity) crossover is observed above a critical concentration of CsPbI3 NCs 
+(Fig. S3 within the Supplimental Material). The possible explanation behind these observations 
+is as follows: upon illumination, photoexcited electron-hole pairs are generated in both WS2 
+and CsPbI3, however, due to the type-II energy band alignment of the heterostructures, 
+electrons are easily transferred from CsPbI3 NCs to ML WS2, as illustrated in Fig. 2(d). On the 
+other hand, photogenerated holes remain trapped in the NCs, resulting in a reduced 
+recombination rate and giving rise to a quenched excitonic PL intensity for CsPbI3 NCs and 
+increased trionic emission in ML WS2. The enhanced generation rate of trions results in the 
+
+reduced density of excitons inside the system, leading to the suppression of excitonic peak 
+intensity and the dominance of the trion peak in the PL spectra of MvWH, as schematically 
+depicted in Fig. 2(e). [34] To further confirm the charge transfer phenomena, time-resolved 
+photoluminescence (Tr-PL) spectra have been measured. Fig. 2(f) shows the Tr-PL decay 
+curves of CsPbI3 NCs (blue curve) and CsPbI3/WS2 MvWHs (red curve). The PL decay curves 
+have been fitted using a bi-exponential function to extract the average excitonic life time (τav). 
+The τav of CsPbI3 NCs decreases from 40.2 ns to 33.7 ns after hybridization with WS2, 
+corroborating the successful charge transfer mechanism from CsPbI3 NCs to WS2 flakes. As a 
+conclusion, the type-II band alignment in CsPbI3/WS2 heterostructure facilitates an efficient 
+electron–hole pair separation and strong electron doping into WS2 channel, making the hybrid 
+system ideal for fabrication of superior performance phototransistor devices [37]. 
+ 
+FIG. 3. (a) Schematic 3D view of the fabricated back gated phototransistor comprising of 
+CsPbI3 sensitized WS2 channel with asymmetric electrodes (Au and Cr) acting as a source and 
+drain. An optical micrograph of the device is shown in the inset. (b) Linear IDS-VDS 
+characteristics of three fabricated devices with different source-drain contacts (i) Cr-Cr 
+(Yellow curve), (ii) Au-Au (Orange curve) and (iii) Au-Cr (Brown curve) without any back 
+gate bias. Inset: the corresponding semi-logarithmic IDS-VDS characteristics plots. (c) Transfer 
+
+(a)
+(b)
+6
+10-
+4
+10-11
+(vu)
+2
+10-13
+T
+Vps (M)
+U
+DS
+-2
+Cr/Cr
+Au
+Au/Au
+4
+Au/Cr
+Watoms
+Si02
+.6
+CsPbl, NCs
+Satoms
+-2
+-1
+0
+1
+2
+V
+Ds (V)
+(c)
+(d)
+10°
+60
+20
+A0
+5V
+10V
+15V
+(vu)
+(vu)
+15
+20V
+'DS
+10-11
+DS
+10
+-20
+20
+40
+20
+WS
+5
+CsPbI,/WS
+4-0
+0
+-20
+0
+20
+40
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+VGs (V)
+V
+(V)
+DS(IDS-VGS) characteristics of the CsPbI3/WS2 hybrid transistor (Green curve) and control WS2 
+transistor (Blue curve) devices in linear scale and logarithmic scale (Inset). The current value 
+in the accumulation region decreases and the threshold voltage is shifted to a higher positive 
+voltage in hybrid device due to charge transfer through the CsPbI3/WS2 junction. (d) Output 
+characteristics of the hybrid device with varying gate voltage.   
+ 
+The CsPbI3 NCs/WS2 MvWH photo-FET with asymmetric metal contacts has been 
+demonstrated by exploiting the Schottky barrier induced dark current suppression for the zero 
+gate bias driven photosensitivity of the device with a simpler fabrication technique. Figure 
+3(a) schematically demonstrates the as-fabricated phototransistor device structure consisting 
+of a few layer WS2 channel and asymmetric Cr and Au electrodes as source and drain terminals, 
+respectively. The inset shows the optical micrograph of the connected few layer WS2 flake 
+(5×2 µm2) having a thickness of ~ 4.5 nm corresponding to 4-5 atomic layers of WS2, further 
+corroborated by the AFM analysis (Fig. S4 within the Supplimental Material). The deposition 
+of a lower work function Cr (ΦCr = 4.5 eV) and higher work function Au (ΦAu = 5.1 eV) on n-
+type WS2 as asymmetric contacts reveals room temperature rectifying diode characteristics 
+with rectification ratio up to 5.2 × 102 even at zero applied back gate bias, as depicted via the 
+current-voltage (IDS-VDS) characteristics in Fig. 3(b). Under an applied reverse drain voltage, 
+the potential barrier height between Au and WS2 becomes higher to suppress the current flow 
+through the junction compared to Cr, and thus exhibits the IDS-VDS characteristics of an ideal 
+diode [13]. The current through the Au–WS2-Cr device follows the diode behaviour and the 
+Schottky barrier (Φb) at Au-WS2 interface is capable of reducing the dark current significantly 
+making the device architecture an ideal prototype for operating in full depletion mode even at 
+zero gate bias, leading to a very high ON-OFF ratio of the device.  On the other hand, a linear 
+IDS-VDS characteristics with a comparatively larger current value (100 nA) confirms the 
+formation of an Ohmic-like junction with a very low contact resistance for Cr-WS2-Cr 
+device [38]. Further, Au-WS2-Au system reveals a rectifying output characteristics with 
+relatively lower current than Cr contacts, confirming a typical high resistive back-to-back 
+
+Schottky diode [15]. The energy band alignment with different metal contacts is schematically 
+depicted in Fig. S5 within the Supplimental Material, revealing an easy current flow through 
+the Ohmic Cr junction and restricted flow via built-in potential barrier in the Au Schottky 
+junction. Fig. 3(c) shows the transfer (IDS-VGS) characteristics of the WS2 phototransistor with 
+asymmetric Cr–Au contacts at a reverse drain voltage of -2V revealing excellent n-type channel 
+properties at room temperature with off-currents of the order of 10 pA and the transistor ON-
+OFF ratio ~104. Further, to understand the effect of CsPbI3 treatment on the device 
+performance, the CsPbI3/WS2 hybrid transistor characteristics is compared with the pristine 
+WS2 one, referred to as the control device. The incorporation of sensitizing perovskite NCs on 
+2D-WS2 layer results in a junction formation via Fermi level alignment in equilibrium under 
+dark condition. In this process, the draining of electrons from the WS2 channel towards CsPbI3 
+NCs results in the depletion of majority carriers in WS2 leading to the lowering of the current 
+flow in the channel under dark condition. Further, we have studied the output characteristics 
+of the hybrid phototransistor on application of gate voltage varying from 0 to +20 V, as depicted 
+in Fig. 3(d). For a higher positive gate voltage, more electrons are induced in the WS2 channel 
+and the transistor has a higher current in the saturation state. On the other hand, under a negative 
+gate bias a small amount of current flows through the channel owing to the depletion of carriers, 
+leading to the OFF state of the transistor [Fig. 3(c)]. 
+The performance of the CsPbI3/WS2 MvWH photo-FET has been analysed by recording the 
+room temperature IDS-VDS characteristics for zero gate bias under dark as well as visible 
+illumination using a Newport solar simulator having broadband emission with irradiance of 
+100 mW/cm2 under air mass (AM) 1.5G condition, as shown in Fig. 4(a). For comparison, the 
+characteristics of the pristine WS2 control device is also presented. The suppression of dark 
+current to the order of tens of pA even without any gate bias along with the significant reduction 
+of noise currents are attributed to the built-in electric field at the Au/WS2 Schottky junction, 
+
+FIG. 4. (a) Comparative IDS-VDS characteristics of pristine WS2 and CsPbI3/WS2 MvWH 
+photo-FET under dark and illumination via broadband light source for zero gate bias. (b) 
+Spectral responsivity curves of MvWH photo-FET at VGS = 0V with increasing reverse VDS 
+from 0V to -2V, as shown via yellow, green and blue curves.  The blue curve represents the 
+responsivity of the photo-FET at a maximum VDS of -2V, while the spectral responsivity of the 
+control device (orange curve) showing an order of magnitude lower device response at same 
+VDS. (c) COMSOL Multiphysics simulated E-field distribution at the vicinity of the hybrid 
+system upon excitation with an excitation wavelength of 680 nm. (d) The transfer 
+characteristics (IDS-VGS) of MvWH photo-FET for a range of incident powers (from 0.1 to 35 
+μW) with an illumination of wavelength 514 nm at VDS = -2 V. (e) The variation of responsivity 
+of the device with incident illumination power for VGS varying from 0 to 40 V. (f) The shift in 
+threshold voltage (ΔVTh) with increasing illumination power (Pin) fitted with a power law. Blue 
+dots represent the extracted data points from panel (a) and green line represents the fitted curve. 
+Inset: The power law fit of the variation of photocurrent with incident power at VDS = -2V and 
+VGS = 40V showing a sublinear photocurrent dependency with incident optical power. 
+ 
+which further helps in effective separation of photogenerated carriers created in WS2 channel. 
+On illuminating the heterojunction device, the reverse current tends to increase due to the 
+collection of photogenerated minority carriers (holes) at the electrodes. The photo-to-dark 
+current ratio of MvWH photo-FETs by illuminating with a broadband light source is estimated 
+to be much higher compared to the pristine WS2 one (~1000 times) under the applied reverse 
+bias condition, which reaches to a value of ~1.08×106 at VDS of -2V, as shown in Fig. S6 within 
+the Supplimental Material. The decoration of WS2 channel with superior light absorbing 
+
+(a)
+(b)
+(c)
+120
+(AW)
+×102
+E2/E.?
+6
+80
+10-8
+4
+MvWHlight
+40
+MvWH dark
+WS,light
+2
+WS,dark
+WS,1
+0
+-2
+-1
+0
+1
+2
+300400500600700800900
+Vps (V)
+Wavelength(nm)
+(d)
+(e)
+(f)
+0.3
+-10
+35μW
+(A/W)
+- 40V
+20V
+200
+10-7
+104
+-15
+(nA)
+40V
+30V
+-10V
+150
+(μA)
+0.2
++
+-OV
+20
+DS
+100
+10-1
+0μw
+Tocp0.48
+10
+50
+10-13
+AV
+-25
+-20
+Vcs (M)
+40
+0
+10
+20
+30
+30
+Pin (μW)
+Resi
+10
+-35
+ocp0.17
+0.0
+40
+-20
+20
+40
+0.1
+1
+10
+0
+7
+14
+0
+21
+28
+35
+(V)perovskite CsPbI3 NCs facilitates enhancement in the photocurrent by elevating the 
+photogenerated carriers in the channel via efficient charge transfer from CsPbI3 to WS2 due to 
+type-II energy band alignment [7,19]. This explains the significant enhancement (~103 times) 
+of response of MvWH transistor over the control one, revealing the role of photoabsorbing 
+CsPbI3 NCs in boosting the performance of the phototransistor. Further, the spectral 
+responsivity of the fabricated MvWH photo-FET, the most important figure of merit to evaluate 
+a detector performance, has been studied displaying a broadband spectral photoresponse 
+covering the entire visible wavelength range, as shown in Fig. 4(b). It may be noted that a peak 
+responsivity of ~1.05×102 A/W at ~ 460 nm at an applied bias (VDS) of -2 V is achieved, which 
+is close to the C-exciton absorption edge of WS2. Two other peaks at ~ 620 nm (R ~ 0.97×102 
+A/W) and ~ 720 nm (R ~ 0.77×102 A/W) correspond to the direct bandgap absorption of few 
+layer WS2 and α-phase CsPbI3 NCs, respectively. Further, the spectral responsivity increases 
+with increasing reverse VDS that assists in the efficient extraction of photogenerated carriers. 
+On the other hand, the control WS2 based device also exhibits a similar trend with increasing 
+bias showing a maximum peak responsivity of ~ 10 A/W at ~ 460 nm at -2 V applied VDS (Fig. 
+S7 within the Supplimental Material). It is to be noted that the decoration of CsPbI3 NCs on 
+WS2 flakes not only improves the detector responsivity by more than 10-fold but also extends 
+the spectral responsivity window up to 800 nm, as shown comparatively in Fig. 4(b). Hence, 
+the decoration of WS2 active channel layer with excellent photoabsorbing CsPbI3 NCs appear 
+to be a promising approach for next generation high performance optoelectronic applications. 
+To further investigate the role of CsPbI3 in photocarrier generation and efficient charge 
+transfer, the electromagnetic simulations have been performed using the COMSOL 
+Multiphysics software. Figure 4(c) shows the electric field distribution of the hybrid 
+CsPbI3/WS2 device, illuminated with an electromagnetic plane wave of wavelength λ = 680 
+nm from top, which propagates through air and the nanostructure. The distribution clearly 
+
+depicts that the electric field is trapped along the edges of the nano-cubes of CsPbI3 with the 
+maximum confinement occurring near the base (as demonstrated by the colour index profile), 
+resulting in strong charge transport in CsPbI3/WS2 hybrid heterostructure.  
+Further to explore the impact of gate bias on transistor performance, the photo-induced transfer 
+characteristics (IDS−VGS) of the WS2/CsPbI3 MvWH photo-FET is recorded under the dark 
+(black markers) and 514 nm illumination with a range of optical powers (from 0.1 μW to 35 
+μW) at a constant VDS of -2V, as illustrated in Fig. 4(d). The corresponding logarithmic current 
+representation is depicted in the inset. Under illumination of a fixed power of 35 μW, the drain 
+current of the MvWH photo-FET is enhanced by ~ 13 times (from ~ 20 nA to ~ 0.26 μA) at a 
+constant gate voltage of ~ 40V, manifested by the strong photoabsorption in CsPbI3 and 
+subsequent transfer of photoexcited electrons to the WS2 channel. Further, with the increase of 
+laser power, the photocurrent significantly increases in the accumulation region (i.e. VGS>VTh) 
+and the transfer curves are gradually shifted to a negative gate voltage. As illustrated in Fig. 
+2(d), the favourable energy band alignment rules out the possibility of hole injection from 
+CsPbI3 into WS2, leading to the trapped holes induced strong photogating effect in the hybrid 
+system. This leads to significant photocurrent increment in the accumulation region and 
+negative threshold voltage shift (ΔVTh) with increasing incident power density of 
+illumination [39]. To investigate in greater detail, the calculated responsivity as a function of 
+illumination power has been plotted for different gate bias voltages in Fig. 4(e).  Here, the 
+responsivity value increases with increasing positive gate bias in case of MvWH photo-FETs 
+and reaches to a high value of ~ 1.1 × 104 A W−1 at a back gate voltage of ~ 40 V under an 
+illumination power of 0.1 μW, which is quite remarkable compared to those previously 
+reported 0D/2D hybrid phototransistors [37,39]. Note that, for all the gate voltages, the 
+measured responsivity dropped with increasing power because of the saturation of sensitizing 
+traps in CsPbI3 NCs, which is a characteristic footprint of trap-dominated photoresponse [40–
+
+42] . Further, we have extracted the threshold voltage via extrapolating the linear region of 
+each transfer curve under different incident laser powers and the shift in threshold voltage is 
+plotted as a function of incident power. The variation is fitted with the power law function 
+𝑉𝑇ℎ ∝ 𝑃𝑏, as depicted in Fig. 4(f), to understand the possible photoconduction mechanism. The 
+extracted fitting exponent, b ~ 0.17 clearly indicates a sublinear dependency on laser power 
+confirming the existence of photogating dominant carrier conduction in MvWH photo-
+FETs [43]. Further, it is also observed that the change in VTh is large in the lower power region 
+and starts to saturate gradually at a higher power owing to the saturated trap states present in 
+sensitizer interface leading to the saturation of the photogating effect. The photocurrent IPh = 
+IPhoto − IDark versus gate voltage for different illumination intensity [Fig. 5(a)] shows a strong 
+modulation with VGS, and a clear maximum in response can be identified around +35 V. The 
+strongest response of the FET device corresponds to the region with highest transconductance, 
+due to the favourable Fermi level alignment, for low-contact resistance operation leading to 
+many cycles of electron circulation to produce maximum gain. Hitherto, in this region the FET 
+device operates at a relatively higher dark current, compromising the signal-to-noise ratio 
+(SNR) of the device, which is also a very important figure of merit of photo-FETs. The SNR 
+defined as IPhoto/IDark is illustrated in the same panel, Fig. 5(a), which reveals the potential of 
+0D/2D hybrid phototransistors for highest sensitivity detection in its depletion regime with VGS 
+from 0 to 5V. In this region, a lowest dark current and a maximum sensitivity are achieved, 
+despite the devices’ concurrent drop in the photocurrent. So the maximum sensitivity of the 
+device can be achieved via contact engineering where the transistor is operated in the depletion 
+region, even without applying any gate bias, hitherto unreported for photo-FET devices. While 
+the peak responsivity of our device is comparable or superior to the reported 2D materials based 
+hybrid phototransistor devices with perovskite sensitizers, the sensitivity is found to be 
+significantly higher without application of any external gate bias (see Table 1). These results 
+
+illustrate the superior performance of broadband phototransistor, with ultrahigh sensitivity and 
+responsivity, using CsPbI3 NCs sensitized 2D WS2 layer. 
+ 
+FIG. 5. (a) Back-gate bias dependent photocurrent (right axis) and photo-to-dark current ratio, 
+(left axis) of the phototransistor device under five different illumination intensities (from 0.1 
+µW to 10 µW) for 514 nm. Despite the strongest photoresponse at higher gate bias (VGS ≈ 40 
+V), highest sensitivity of the device is achieved in the depletion regime (VGS ≈ 0V). The 
+schematic representation of channel current transport mechanism and energy band diagram of 
+the asymmetric contact hybrid phototransistor under reverse drain-source voltage with (b) zero 
+and (c-d) different gate bias conditions.  
+ 
+On the other hand, a remarkable photoresponse of CsPbI3/WS2 MvWH photo-FET is explained 
+by considering the influence of positive gate voltage on energy band alignment at the contact 
+interfaces and heterostuctures leading to efficient charge injection into n-type WS2 channel  
+Table 1. Comparison of device performances with reported 2D material based hybrid photo-
+FETs with perovskite sensitizers 
+
+(a)
+(b)
+hy
+High sen sitivity
+High photoresponse
+104
+90
+0.1 μW
+0.5 μW
+(vu)
+5 μW
+10μW
+CsPbI3
+103
+0.00
+60
+Photo
+10
+Au
+WS2
+Cr
+30
+10
+-ve
++ve
+100
+00
+20
+0
+20
+40
+Vcs (V)
+Underillumination
+(c)
+(p)
+hy
+CsPbl
+CsPbI3
+Au
+Au
+Cr
+WS2
+-ve
+-ve
+Cr
+WS2
++ve
++ve
+Depletion region
+Accumulation regionDevice 
+structure 
+Sensitizer 
+Operational 
+spectral 
+range 
+Idark 
+w/o 
+applied 
+VGS 
+Iphoto/Idark 
+@ 
+VGS=0V 
+Responsivity 
+for different 
+values of VGs 
+Ref. 
+Au / ML 
+WS2 / Au 
+CH3NH3PbI3 
+450-700 nm 
+5nA 
+104 
+2.5 A/W @ 0V   [44] 
+Au / ML 
+MoS2 / Au 
+CsPbBr3 
+350-550nm 
+0.2 nA 
+103 
+4.4 A/W @ 0V   [45] 
+Au / Ti / ML 
+MoS2 / Ti / 
+Au 
+Ch3NH3PbBr3 
+/ CsPbI3-xBrx 
+532 and 355 
+nm 
+4 nA 
+104 
+7 × 104 A/W 
+@ 60V  
+ [46] 
+Au / Ti / FL 
+MoS2 / Ti / 
+Au 
+CsPbBr3 
+405 nm 
+10 nA 
+10 
+4.7 × 104 A/W 
+@ 20V  
+ [47] 
+Au / FL BP / 
+Au 
+CsPbBr3 
+405 nm 
+2 nA 
+102 
+357.2 mA/W 
+@ 0V  
+ [48] 
+Au / ML 
+MoS2 / Au 
+CsPbI3-xBrx 
+532 nm 
+0.2 µA 
+103 
+1.13 × 105 
+A/W @ 60V  
+ [49] 
+Au / FL BP / 
+Al / Au 
+MAPbI3−xClx 
+400-900 nm 
+0.1 µA 
+/µm 
+102 
+4 × 106 A/W 
+@ 40V  
+ [50] 
+Au / Ti / FL 
+MoSe2(WSe2
+) / Ti / Au 
+CsPb(Cl/Br)3 
+455 nm 
+0.8 nA 
+10 
+102 A/W @ 
+50V  
+ [51] 
+Au / Cr / FL 
+Ta2NiSe5 / Cr 
+/ Au 
+CH3NH3PbI3 
+800 nm 
+3.5 µA 
+10 
+2.4 × 102 A/W 
+@ 0V  
+ [37] 
+Au / Cr / FL 
+WS2 / Au / 
+Cr 
+α-phase 
+CsPbI3 
+400-800 nm 
+2 pA 
+106 
+104 A/W @ 
+40V  
+Our 
+work 
+ 
+layer from photoabsorbing CsPbI3 NCs. As illustrated in Fig. 5(b), the Schottky barrier at the 
+Au/WS2 interface is high enough to inhibit the charge conduction mechanism across the WS2 
+channel layer at reverse drain bias without any gate electric field under dark condition. Hence, 
+the transistor immediately goes to the OFF state with very low dark current in the order of pA. 
+At this condition, when the visible light is illuminated on the 0D/2D heterostructure, the 
+photogeneration takes place in both CsPbI3 NCs as well as WS2 channel layer, as depicted in 
+Fig. 5(b). The effective photogenerated carrier separation takes place by the built-in electric 
+field at the Schottky junction (WS2/Au) as well as at CsPbI3/WS2 interfaces. The subsequent 
+transition of photoexcited electrons from CsPbI3 to WS2 starts to populate the active channel 
+
+layer which are collected by the external electrodes under an applied reverse VDS, leading to 
+the photoresponsivity of ~ 102 A/W at VDS = ‒2V and VGS = 0V. Further, the application of a 
+back gate voltage (VGS) to the device modulates the Schottky barrier height at Au/WS2 interface 
+as shown in Figs. 5(c)-(d) [52,53]. An application of negative gate bias (VGS < VTh) increases 
+the barrier height leading to the transistor operation in the depletion region, where the 
+photosensitivity (IPhoto/IDark) of the device is maximum. On the other hand, on increasing the 
+VGS beyond VTh initiates the lowering of the Schottky barrier at Au/WS2 interface, resulting in 
+a higher magnitude of charge carrier injection from the Au electrode to WS2 channel layer 
+through thermoionic as well as tunnelling mechanisms, as illustrated in Fig. 5(d). Thus, the 
+cumulative effects of CsPbI3 NCs decoration mediated strong photogating phenomena as well 
+as the gate voltage induced Schottky barrier lowering result in a drastic enhancement of the 
+photocurrent (IPhoto − IDark) through the transistor channel at ON state (VGS > VTh). This leads 
+to an ultrahigh photoresponsivity of the order of ~ 104 A/W at VGS = 40 V. Such gate modulated 
+responsivity and sensitivity of MvWH photo-FET devices via interface engineering offers a 
+novel pathway for next generation high performance and low power integrated photonic 
+technology.   
+Temporal photoresponse is also an important parameter for the phototransistors performance 
+in terms of switching speed and device stability. The transient photoresponse of the as-
+fabricated CsPbI3/WS2 MvWH photo-FET upon visible illumination (𝜆 = 514 nm) at VGS = 0V 
+with varying reverse VDS is demonstrated in Fig. 6(a). Upon illumination of four periodic 
+pulses of the Argon-ion laser, relatively fast and consistent photocurrent modulation 
+characteristics of the device reveals the stability and reproducibility of the as-fabricated MvWH 
+photo-FET. The device exhibits a much stronger photoresponse characteristics revealing ratio 
+of ~106 as compared to the control device with pristine WS2 with the value ~103 (Fig. S8 within 
+the Supplimental Material), which is attributed to the injection of high density 
+
+ 
+FIG. 6. (a) Transient response of the MvWH photo-FET device under illumination of a 514 nm 
+laser at different applied VDS. (b) Temporal photocurrent response of the MvWH device for a 
+wavelength of 514 nm with and without any applied gate bias. The temporal response indicates 
+a significant decrease in rise time (from 43.8 to 34 ms) as well as fall time (from 32.7 to 24 
+ms), measured at a relatively higher power of 35 µW. (c) Operational stability of the fabricated 
+MvWH photo-FET device under visible illumination for more than half an hour. (d) The 
+transient photocurrent response of the fabricated transistor over a span of seven days from the 
+beginning and end of the stability test. (e) Stability of the device tested under extreme humid 
+conditions (varying from 50 to 65% RH). The last four cycle is the response under 65% of 
+humidity showing around 5% decay in the photoresponse.  
+ 
+ photogenerated charge carriers into the WS2 channel from strong light absorbing CsPbI3 NCs. 
+With the increment of reverse VDS, a consistent photocurrent enhancement is distinctly noticed 
+from the switching characteristics owing to the increase of depletion region width at the 
+Schottky barrier interface and subsequent separation of photogenerated charge carriers. 
+Further, the rise and fall times of the fabricated device in the absence of gate bias have been 
+estimated using an enlarged single cycle response [Fig. 6(b)] and are found to be around ∼ 
+43.8 ms and ∼ 32.7 ms, respectively, which are further reduced to 34 ms and 24 ms, 
+respectively on applying a gate voltage of 40 V. The response speed of these devices are found 
+to be relatively slower, which is attributed to the trapping of charge carriers in various structural 
+
+(a)
+(c)
+3
+21
+-1.5V
+-1V
+2.5
+2.0
+HA
+1.5
+DS
+DS
+1.0
+0.5
+ON
+ON
+0
+OFF
+0.0
+0
+20
+40
+60
+80
+0
+20
+40
+0
+602
+1205
+1807
+18901920
+Time (sec)
+Time (sec)
+Time (sec)
+Time (sec)
+(b)
+(d)
+(e)
+1.2
+1.2
+oV
+3
+VDs = -2V
+2=514nm
+50% humidity
+65% humidity
+GS
+Dav 1
+Day3
+Day 5
+Dav 7
+Normalized 
+0.8
+Normalized
+0.8
+(vn)
+2
+0.4
+DS
+0.4
+0.0
+0.0
+5%
+decay
+6.75
+6.80
+6.85
+6.90
+6.95
+7.00
+Time (sec)
+Time (sec)
+Time (sec)and surface defect states present in WS2 as well as CsPbI3 NCs and their local junction 
+interfaces. These interface traps present in WS2 layer are mostly empty when biased under 
+depletion condition, i.e. VGS < VTh owing to lack of enough mobile carriers in the channel. This 
+allows a large number of photogenerated electrons to get trapped by the defect states while 
+some of the gate induced electrons, although small in number, can be trapped as well. This 
+results in a relatively slow rise of current, as depicted in the photocurrent dynamic response. 
+On the other hand, the interface traps are nearly filled up with gate-induced electrons in 
+accumulation condition, when VGS > VTh, as shown in Fig. 6(b). Hence, the trapping 
+probability of photogenerated carriers is lower and a relatively faster response (~34 ms) is 
+observed in MvWH photo-FET devices. Further, owing to the fact that the perovskite materials 
+are prone to environmental degradation via oxygen diffusion through iodide vacancies upon 
+illumination, the long term operation stability of the fabricated devices have been tested in this 
+study upon visible light illumination at zero gate bias for prolonged duration (more than 60 
+min). From the I–t curves for the first 100 s [Fig. 6(c), left] and the last 100 s [Fig. 6(c), right], 
+it is observed that the photocurrent has almost no attenuation, indicating that these devices 
+show an excellent light stability under ambient condition, even without the use of a glovebox 
+or encapsulation. The device stability has also been tested via recording the photocurrent under 
+illumination over a period of one week, as illustrated in Fig. 6(d). Here, the phototransistor 
+sustains under laboratory ambient conditions (relative humidity (RH) ~ 45-50%, temperature 
+~ 22oC) for one week with negligible change in the photocurrent via degradation after storing. 
+Further, as CsPbI3 NCs are vulnerable to environmental humidity, to explore the device 
+performance in the extreme humid condition, we have performed the temporal response under 
+65% RH showing an insignificant degradation (5% decay) in terms of device response [Fig. 
+6(e)].  This superior performance stability is due to the surface defect passivation of CsPbI3 
+through the interaction with the sulphur of WS2 ensuring the outstanding environmental 
+
+stability of as-fabricated CsPbI3/WS2 MvWH photo-FETs. The sulfur atoms present on the top 
+layer of WS2 may have stronger coordination to the Pb2+ centers of CsPbI3 NCs leads to reduced 
+defect states in perovskites enabling higher reluctance to the degradation  [31]. It may be noted 
+that the performance of the devices could be further improved by process optimization, device 
+encapsulation and incorporation of buffer layers. This work reveals the significant potential of 
+colloidal synthesized air-stable α-CsPbI3 NCs on 2D materials in fabricating 0D/2D mixed-
+dimensional heterostructure photo-FETs for applications in next generation optoelectronic 
+devices.  
+Conclusion: 
+To summarize, significant improvements in performance have been realized in CsPbI3/WS2 
+0D/2D mixed-dimensional phototransistors with asymmetric metal electrodes leading to 
+combinatorial effect of Schottky barrier induced suppression of dark current and efficient 
+charge transfer from photoabsorbing CsPbI3 nanocrystals, resulting in enhanced 
+photosensitivity and spectral responsivity. The WS2 channel with asymmetric contacts 
+(Cr/WS2/Au) shows a rectifying I-V characteristics under an applied VDS with the dark current 
+in the order of pA. Further, by combining the channel sensitization via decorating the WS2 with 
+photosensitive air-stable α-phase CsPbI3 NCs, a responsivity of ~102 A/W has been achieved 
+at low VDS (~ -2V) for an incident optical power of 0.1 µW even without any external gate 
+bias. The device exhibits a broad spectral photoresponsivity between 400 and 800 nm due to 
+the extended visible light absorption features of CsPbI3 NCs. Using gate-controlled carrier 
+modulation in the transistor channel, a peak responsivity ~104 A/W (VGS = +40 V) has been 
+achieved owing to the photogating effect mediated charge conduction whereas the maximum 
+sensitivity (~ 106 at ~ VDS = -2 V) in terms of signal-to-noise ratio is observed by depleting the 
+channel carries (VGS = 0 to 5 V). These devices show superior performance in terms of 
+environment stability, owing to the filling of surface trap states present in CsPbI3 NCs via 
+
+conjugation with sulfur atoms of 2D WS2 layer. The fabricated hybrid heterostructure devices 
+combining 2D TMDs and superior light absorbing 0D perovskite nanocrsytals, through proper 
+interface engineering, would open up new pathways for novel optoelectronic functionalities 
+and energy-harvesting applications. 
+Acknowledgement:  
+SKR acknowledges the support of Chair Professor Fellowship of the Indian National Academy 
+of Engineering (INAE). 
+Conflicts of interest 
+There are no conflicts to declare. 
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+ 
+ 
+ 
+
+Supplemental Material 
+ 
+Charge transfer mediated giant photo-amplification in air-stable α-CsPbI3 
+nanocrystals decorated 2D-WS2 photo-FET with asymmetric contacts 
+Shreyasi Das1, Arup Ghorai1,2, Sourabh Pal3, Somnath Mahato1, Soumen Das4, Samit K. Ray5 * 
+1School of Nano Science and Technology, IIT Kharagpur, Kharagpur, West Bengal, India, 721302 
+2Department of Materials Science and Engineering, Pohang University of Science and Technology, 
+Pohang 790-784, Korea 
+3Advanced Technology Development Centre, IIT Kharagpur, Kharagpur, West Bengal, India, 721302 
+4School of Medical Science and Technology, IIT Kharagpur, Kharagpur, West Bengal, India, 721302 
+5Department of Physics, IIT Kharagpur, Kharagpur, West Bengal, India, 721302 
+Email : physkr@phy.iitkgp.ac.in 
+ 
+ 
+ 
+
+S1: Tauc plot of CsPbI3 NCs 
+ 
+Fig. S1. Tauc plot of as synthesised α-phase CsPbI3 NCs  
+ 
+S2: Photoluminescence spectra of WS2, CsPbI3 and CsPbI3/WS2 hybrid 
+ 
+Fig. S2. Comparative PL spectrum of bare ML WS2 flake, CsPbI3 NCs and their 
+heterostructures where three different concentrations of CsPbI3 NCs (different steps of spin 
+coating) incorporated on WS2 flakes. After formation of heterostructures, the PL intensity of 
+both bare ML WS2 as well as CsPbI3 NCs get reduced. 
+
+(αhv)? (a.u.)
+E. = 1.814 eV
+1.6
+1.8
+2.0
+2.2
+2.4
+2.6
+Energy (eV)PL Intensity (a.u.)
+Ws?
+CsPbI3
+Step 1
+Step 2
+Step 3
+550
+600
+650
+700
+750
+Wavelength (nm)S3: PL integrated intensity ratio vs coating step 
+ 
+Fig. S3. Variation of PL integrated intensity ratio of WS2 trion peak (A-) to excitonic peak 
+(A) with increasing concentration of CsPbI3 decoration (increasing spin coating step) on WS2 
+flakes. 
+ 
+S4: Thickness of the exfoliated flake analysis using AFM  
+ 
+Fig. S4. Atomic force microscopy image of the few layer WS2 flakes. Inset shows the height 
+profile along the yellow dashed line confirming the thickness of the flakes around 4.5 nm.  
+
+7
+Increasing CsPbI,
+6
+concentration
+5
+2
+1
+0
+Step 3
+Step 2
+Step 1
+Ws.0
+5
+10
+15
+20
+25
+30
+35
+40
+45μm
+nm
+30
+Height (nm)
+5
+27.5
+25
+10-
+22.5
+15
+-20
+20
+0
+17.5
+25
+0.0
+0.4
+0.8
+1.2
+-15
+Distance (um)
+30-
+12.5
+-10
+35
+7.5
+40
+-5
+45
+2.5
+50
+umS5: Energy band structures of WS2 at the contacts 
+ 
+Fig. S5. The corresponding energy band structures with different combination of metal 
+electrodes before contact and after contact condition under applied reverse bias. 
+ 
+S6: Photo to dark current ratio 
+ 
+Fig. S6. Photo to dark current ratio for control device (WS2 FET) and MvWH photo-FET 
+with varying drain to source voltage. 
+Cr
+Cr
+WS2
+Ohmic contact
++ve
+-ve
+Au
+Au
+WS2
+Symmetric contact
++ve
+-ve
+Au
+-ve
+Cr
++ve
+WS2
+Asymmetric contact
+Schottky diode
+Au
+Cr
+WS2
+4.6 eV
+4.5 eV
+5.1 eV
+Evac
+EF
+Before contact
+
+1.2x10
+Ws
+9.0x1(
+sPbI.
+rk
+Dal
+6.0x10
+3.0x10
+0.0
+2.0
+-1.5
+-1.0
+-0.5
+0.0
+Vps (V)S7: Spectral responsivity of the control WS2 FET device  
+ 
+Fig. S7. Spectral responsivity curves of control WS2 FET device at VGS = 0V with increasing 
+reverse VDS from 0V to -2V 
+ 
+S8: Transient response of the control WS2 FET device 
+ 
+ 
+Fig. S8. Transient response of the control WS2 FET device under illumination of a 514 nm 
+laser at different applied VDS. 
+ 
+300 400 500 600 700 800 900
+0
+3
+5
+8
+10
+Responsivity (A/W)
+WS2
+Wavelength (nm)
+ -2V
+ -1V
+ 0V
+
+8
+-1.5V
+-1V
+40
+6
+(nA)
+4
+2
+ON
+OFF ON
+0
+0
+20
+40
+60
+80
+Time (sec)

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+On the structure of entropy solutions to the Riemann
+problem for a degenerate nonlinear parabolic equation
+Evgeny Yu. Panov
+Yaroslav-the-Wise Novgorod State University, Veliky Novgorod, Russian Federation,
+Research and Development Center, Veliky Novgorod, Russian Federation.
+Abstract
+We find an explicit form of entropy solutions to a Riemann problem for a de-
+generate nonlinear parabolic equation with piecewise constant velocity and diffusion
+coefficients. It is demonstrated that this solution corresponds to the minimum point
+of some strictly convex function of a finite number of variables.
+1
+Introduction
+In a half-plane Π = {(t, x) | t > 0, x ∈ R}, we consider a nonlinear parabolic equation
+ut + v(u)ux − t(a2(u)ux)x = 0,
+(1)
+where v(u), a(u) ∈ L∞(R), a(u) ≥ 0. Since the diffusion coefficient a(u) may take zero
+value, equation (1) is degenerate. In the case when a(u) ≡ 0 it reduces to a first order
+conservation law
+ut + ϕ(u)x = 0,
+(2)
+where ϕ′(u) = v(u). Similarly, a general equation (1) can be written in the conservative
+form
+ut + ϕ(u)x − tA(u)xx = 0
+with A′(u) = a2(u), which allows to define weak solutions of this equation. Unfortunately,
+weak solutions to a Cauchy problem for equation (1) are not unique in general, and some
+additional entropy conditions are required. We consider the Cauchy problem with initial
+data
+u(0, x) = u0(x),
+(3)
+where u0(x) ∈ L∞(R). Recall the notion of entropy solution (e.s. for short) in the sense of
+Carrillo [1].
+Definition 1. A function u = u(t, x) ∈ L∞(Π) is called an e.s. of (1), (3) if
+(i) the distribution A(u) ∈ L2
+loc(Π);
+(ii) for all k ∈ R
+|u − k|t + (sign(u − k)(ϕ(u) − ϕ(k)))x − (t sign(u − k)(A(u) − A(k)))xx ≤ 0
+(4)
+in the sense of distributions (in D′(Π));
+(iii) ess lim
+t→0
+u(t, ·) = u0 in L1
+loc(R).
+1
+arXiv:2301.13292v1  [math.AP]  30 Jan 2023
+
+Entropy condition (4) means that for each nonnegative test function f = f(t, x) ∈
+C2
+0(Π)
+�
+Π
+[|u − k|ft + sign(u − k)((ϕ(u) − ϕ(k))fx + t(A(u) − A(k))fxx)]dtdx ≥ 0.
+(5)
+In the case of conservation laws (2) the notion of e.s. reduces to the notion of generalized
+e.s. in the sense of Kruzhkov [2]. Taking in (4) k = ±M, M ≥ ∥u∥∞, we derive that
+ut + ϕ(u)x − tA(u)xx = 0 in D′(Π),
+that is, an e.s. u of (1), (3) is a weak solution of this problem. It is known that an e.s. of (1),
+(3) always exists and is unique. In general multidimensional setting this was demonstrated
+in [2] for conservation laws and in [1] for the general case. If to be precise, in [1] the case of
+usual diffusion term A(u)xx was studied but the proofs can be readily adapted to the case
+of the self-similar diffusion tA(u)xx.
+If u = u(t, x) is a piecewise C2-smooth e.s. of equation (1) then it must satisfy this
+equation in classic sense in each smoothness domain.
+Applying relation (1) to a test
+function f = f(t, x) ∈ C2
+0(Π) supported in a neighborhood of a discontinuity line x = x(t)
+and integrating by parts, we then obtain the identity
+(−x′(t)[u] + [ϕ(u)] − t[A(u)x])f + t[A(u)]fx = 0
+(6)
+a.e. on the line x = x(t). Here we denote by [w] the jump of a function w = w(t, x) on the
+line x = x(t) so that
+[w] = w(t, x(t)+) − w(t, x(t)−),
+where w(t, x(t)±) =
+lim
+y→x(t)± w(t, y).
+Since the functions f, fx are arbitrary and independent on the line x = x(t), identity (6)
+implies the following two relations of Rankine-Hugoniot type
+[A(u)] = 0,
+(7)
+−x′(t)[u] + [ϕ(u)] − t[A(u)x] = 0.
+(8)
+Similarly, it follows from entropy relation (5), after integration by parts, that
+(−x′(t)[|u − k|] + [sign(u − k)(ϕ(u) − ϕ(k))] − t[sign(u − k)A(u)x])f+
+t[sign(u − k)(A(u) − A(k))]fx ≤ 0.
+(9)
+Since the function A(u) increases, it follows from (7) that A(u) = const when u lies between
+the values u(t, x(t)−) and u(t, x(t)+). This implies that [sign(u − k)(A(u) − A(k))] = 0
+and in view of arbitrariness of f ≥ 0 it follows from (9) that
+− x′(t)[|u − k|] + [sign(u − k)(ϕ(u) − ϕ(k))] − t[sign(u − k)A(u)x] ≤ 0.
+(10)
+In the case when k lies out of the interval with the endpoints u± .= u(t, x(t)±) relation (10)
+follows from (8) and fulfils with equality sign. When u− < k < u+ this relation reads
+−x′(t)(u+ + u− − 2k) + ϕ(u+) + ϕ(u−) − 2ϕ(k) − t(A(u)+
+x + A(u)−
+x ) ≤ 0,
+2
+
+where A(u)±
+x = A(u)x(t, x(t)±). Adding (8) to this relation and dividing the result by 2,
+we arrive at the following analogue of the famous Oleinik condition (see [3]) known for
+conservation laws.
+− x′(t)(u+ − k) + ϕ(u+) − tA(u)+
+x − ϕ(k) ≤ 0
+∀k ∈ [u−, u+].
+(11)
+In the case u+ < u− this condition has the form
+− x′(t)(u+ − k) + ϕ(u+) − tA(u)+
+x − ϕ(k) ≥ 0
+∀k ∈ [u+, u−]
+(12)
+and can be derived similarly.
+Geometric interpretation of these conditions is that the
+graph of the flux function ϕ(u) lies not below (not above) of the segment connecting the
+points (u−, ϕ(u−) − tA(u)−
+x ), (u+, ϕ(u+) − tA(u)+
+x ) when u− ≤ u ≤ u+ (respectively, when
+u+ ≤ u ≤ u−), see Figure 1. We take here into account that in view of condition (8)
+the vector (−x′(t), 1) is a normal to the indicated segment. We also notice that it follows
+from relations (11), (12) with k = u± and from the Rankine-Hugoniot condition (8) that
+A(u)±
+x ≥ 0 (A(u)±
+x ≤ 0) whenever u+ > u− (u+ < u−).
+u
+u-
+u+
+y=φ(u)
+y
+tA(u)x
+-
+tA(u)x
++
+(-x',1)
+k
+Figure 1: Oleinik condition.
+2
+The case of piecewise constant coefficients.
+Below we will assume that the functions v(u), a(u) are piecewise constant, v(u) = vk,
+a(u) = ak when uk < u < uk+1, k = 0, . . . , n − 1, where
+α = u0 < u1 < · · · < un−1 < un = β.
+We will study problem (1), (3) with the Riemann data u0(x) =
+� α,
+x < 0,
+β,
+x > 0.
+Since this
+problem is invariant under the scaling transformations t → λt, x → λx, λ > 0 then, by
+the uniqueness, the e.s. u = u(t, x) is self-similar: u(t, x) = u(λt, λx). This implies that
+u = u(x/t). Suppose that ak > 0. Then in a domain where uk < u(ξ) < uk+1 with ξ = x/t
+equation (1) reduces to the second order ODE
+(vk − ξ)u′ − a2
+ku′′ = 0,
+3
+
+the general solution of which is u = C1F((ξ − vk)/ak) + C2; C1, C2 = const, where
+F(z) =
+1
+√
+2π
+� z
+−∞
+e−s2/2ds
+is the error function. Therefore, it is natural to seek the e.s. of our problem in the following
+form
+u(ξ) =
+� uk +
+uk+1−uk
+F((ξk+1−vk)/ak)−F((ξk−vk)/ak)(F((ξ − vk)/ak) − F((ξk − vk)/ak))
+,
+ak > 0,
+uk
+,
+ak = 0,
+(13)
+ξk < ξ < ξk+1, k = 0, . . . , d,
+where
+d =
+� n − 1
+,
+an−1 > 0,
+n
+,
+an−1 = 0,
+− ∞ = ξ0 < ξ1 ≤ · · · ≤ ξd < ξd+1 = +∞,
+and we agree that an = 0, F(−∞) = 0, F(+∞) = 1. We also assume that ξk+1 > ξk
+whenever ak > 0. The rays x = ξkt for finite ξk are (weak or strong) discontinuity lines of
+u, they correspond to discontinuity points ξk of the function u(ξ). Observe that conditions
+(7), (8) turns into the following relations at points ξk
+[A(u)] = A(u(ξk+)) − A(u(ξk−)) = 0,
+(14)
+−ξk[u] + [ϕ(u)] − [A(u)′] = −ξk(u(ξk+) − u(ξk−)) + ϕ(u(ξk+))−
+ϕ(u(ξk−)) − A(u)′(ξk+) + A(u)′(ξk−) = 0.
+(15)
+Here w(ξk±) denotes unilateral limits of a function w(ξ) at the point ξk. Similarly, the
+Oleinik condition (11) reads
+− ξk(u(ξk+) − k) + ϕ(u(ξk+)) − A(u)′(ξk+) − ϕ(k) ≤ 0
+∀k ∈ [u(ξk−), u(ξk+)].
+(16)
+Notice that our solution (13) is a nonstrictly increasing function of the self-similar variable
+ξ and, therefore, u(ξk−) ≤ u(ξk+).
+Let us firstly analyze the solution (13) in the case ξk−1 < ξk < ξk+1. If ak−1, ak > 0
+then u(ξk−) = u(ξk+) = uk so that condition (14) fulfils while (15) reduces to the equality
+[A(u)′] = 0, which is revealed as
+ak(uk+1 − uk)
+F((ξk+1 − vk)/ak) − F((ξk − vk)/ak)F ′((ξk − vk)/ak) =
+ak−1(uk − uk−1)
+F((ξk − vk−1)/ak−1) − F((ξk−1 − vk−1)/ak−1)F ′((ξk − vk−1)/ak−1).
+(17)
+In this situation ξ = ξk is a weak discontinuity point, the function u(ξ) itself is continuous,
+only its derivative u′(ξ) may be discontinuous. Moreover, it follows from (17) that both
+functions u(ξ) and u′(ξ) are continuous at point ξk if ak = ak−1 > 0.
+If ak−1 > ak = 0 then again u(ξ) is continuous at uk and (15) reduces to the relation
+ak−1(uk − uk−1)
+F((ξk − vk−1)/ak−1) − F((ξk−1 − vk−1)/ak−1)F ′((ξk − vk−1)/ak−1) = 0,
+(18)
+4
+
+which is impossible. If ak > ak−1 = 0 then u(ξk−) = uk−1 < uk = u(ξk+), that is, ξk is
+a strong discontinuity point. Condition (14) holds because A(u) is constant on [uk−1, uk]
+(A′(u) = a2
+k−1 = 0) while (15) turns into
+(vk−1 − ξk)(uk − uk−1) −
+ak(uk+1 − uk)
+F((ξk+1 − vk)/ak) − F((ξk − vk)/ak)F ′((ξk − vk)/ak) = 0, (19)
+where we use the fact that ϕ(uk) − ϕ(uk−1) = vk−1(uk − uk−1). It remains to analyze the
+situation when ak = ak−1 = 0. In this case again u(ξk−) = uk−1 < uk = u(ξk+) and
+A(uk−1) = A(uk) while condition (15) turns into the simple relation
+ξk = vk−1.
+(20)
+Finally, since the function ϕ(u) is affine on the segment [uk−1, uk] and A(u)′(ξk±) ≥ 0, then
+entropy relation (16) is always satisfied.
+Now we consider the case when there exists a nontrivial family of mutually equaled
+values ξi, ξi = ξk for i = k, . . . , l, where l > k. We can assume that this family is maximal,
+that is,
+−∞ ≤ ξk−1 < ξk = · · · = ξl < ξl+1 ≤ +∞.
+Then ai = 0 for i = k, . . . , l − 1, and the point ξ = c .= ξk is a discontinuity point of u(ξ)
+with the unilateral limits
+u(c+) = ul, u(c−) = uk′,
+where k′ =
+� k
+,
+ak−1 > 0,
+k − 1
+,
+ak−1 = 0 .
+Since a(u) = 0 for u(c−) < u < u(c+), we find that A(u(c−)) = A(u(c+)) and condition
+(14) is satisfied. Further, we notice that
+l−1
+�
+i=k′
+(−ξi+1(ui+1 − ui)) = −c
+l−1
+�
+i=k′
+(ui+1 − ui) = −c(ul − uk���),
+l−1
+�
+i=k′
+vi(ui+1 − ui) =
+l−1
+�
+i=k′
+(ϕ(ui+1) − ϕ(ui)) = ϕ(ul) − ϕ(uk′).
+Therefore, condition (15) can be written in the form
+l−1
+�
+i=k′
+(vi − ξi+1)(ui+1 − ui) − (A(u)′(c+) − A(u)′(c−)) = 0,
+(21)
+where, as is easy to verify,
+A(u)′(c−) =
+�
+0
+,
+ak−1 = 0,
+ak−1(uk−uk−1)
+F((ξk−vk−1)/ak−1)−F((ξk−1−vk−1)/ak−1)F ′((ξk − vk−1)/ak−1)
+,
+ak−1 > 0,
+(22)
+A(u)′(c+) =
+�
+0
+,
+al = 0,
+al(ul+1−ul)
+F((ξl+1−vl)/al)−F((ξl−vl)/al)F ′((ξl − vl)/al)
+,
+al > 0.
+(23)
+5
+
+In the similar way we can write the Oleinik condition (16) as follows
+l−1
+�
+i=j
+(vi − ξi+1)(ui+1 − ui) − A(u)′(c+) ≤ 0
+k′ < j < l.
+(24)
+We use here the fact the function ϕ(u) is piecewise affine and, therefore, it is enough to
+verify the Oleinik condition (16) only at the nodal points k = uj.
+The above reasoning remains valid also in the case when l = k. In this case, relation
+(21) reduces to one of conditions (17), (18), (19), (20) while (24) is trivial.
+3
+The entropy function
+We introduce the convex cone Ω ⊂ Rd consisting of points ¯ξ = (ξ1, . . . , ξd) with increasing
+coordinates, ξ1 ≤ ξ2 ≤ · · · ≤ ξd such that ξk+1 > ξk whenever ak > 0, k = 1, . . . , d − 1.
+Each point ¯ξ ∈ Ω determines a function u(ξ) in correspondence with formula (13). Assume
+firstly that ¯ξ ∈ Int Ω, that is, the values ξk are strictly increasing. Then conditions (17),
+(18), (19), (20) coincides with the equality
+∂
+∂ξk E(¯ξ) = 0, where
+E(¯ξ) = −
+�
+k=0,...,n−1,ak>0
+(ak)2(uk+1 − uk) ln(F((ξk+1 − vk)/ak) − F((ξk − vk)/ak))+
+1
+2
+�
+k=0,...,n−1,ak=0
+(uk+1 − uk)(ξk+1 − vk)2,
+¯ξ = (ξ1, . . . , ξd) ∈ Ω.
+(25)
+We will call this function the entropy because it depends only on the discontinuities of a
+solution. Thus, for ¯ξ ∈ Int Ω the e.s. (13) corresponds to a critical point of the entropy. We
+are going to demonstrate that the entropy is strictly convex and coercive in Ω. Therefore,
+it has a unique global minimum point in Ω. In the case when this minimum point lies in
+Int Ω it is a unique critical point.
+Obviously, E(¯ξ) ∈ C∞(Ω). Notice that for all k = 0, . . . , n − 1, such that ak > 0
+ln(F((ξk+1 − vk)/ak) − F((ξk − vk)/ak)) < 0.
+Therefore, all terms in expression (25) are nonnegative and, in particular, E(¯ξ) ≥ 0.
+Proposition 1 (coercivity). The sets E(¯ξ) ≤ c are compact for each constant c ≥ 0.
+Proof. If E(¯ξ) ≤ c then it follows from nonnegativity of all terms in (25) that for all
+k = 0, . . . , n − 1
+−(ak)2(uk+1 − uk) ln(F((ξk+1 − vk)/ak) − F((ξk − vk)/ak)) ≤ E(¯ξ) ≤ c if ak > 0,
+(26)
+(uk+1 − uk)(ξk+1 − vk)2/2 ≤ c if ak = 0.
+(27)
+Relation (26) implies the estimate
+F((ξk+1 − vk)/ak) − F((ξk − vk)/ak) ≥ δ .= exp(−c/m) > 0,
+(28)
+where m =
+min
+k=0,...,n−1,ak>0(ak)2(uk+1 − uk) > 0. If a0 > 0 relation (28) with k = 0 reads
+F((ξ1 − v0)/a0) > δ (notice that F((ξ0 − v0)/a0) = F(−∞) = 0), which implies that
+ξ1 ≥ v0 + a0F −1(δ).
+6
+
+On the other hand, if a0 = 0 then (u1 − u0)(ξ1 − v0)2 ≤ 2c, in view of (27) with k = 0, and
+ξ1 ≥ v0 − (2c/(u1 − u0))1/2.
+In any case,
+ξ1 ≥ r1 .= v0 + min(a0F −1(δ), −(2c/(u1 − u0))1/2).
+(29)
+To get an upper bound, we remark that in the case an−1 > 0 it follows from (28) with
+k = d = n − 1 that F(−(ξn−1 − vn−1)/an−1) = 1 − F((ξn−1 − vn−1)/an−1) ≥ δ (observe that
+F((ξn − vn−1)/an−1) = F(+∞) = 1), which implies the estimate
+ξd ≤ vn−1 − an−1F −1(δ).
+If an−1 = 0 then d = n and in view of inequality (27) with k = n − 1 we find (un −
+un−1)(ξn − vn−1)2/2 ≤ c, that is,
+ξd ≤ vn−1 + (2c/(un − un−1))1/2.
+In both cases
+ξd ≤ r2 .= vn−1 + max(−an−1F −1(δ), (2c/(un − un−1))1/2).
+(30)
+Since all coordinates of ¯ξ lie between ξ1 and ξd, estimates (29), (30) imply the bound
+|¯ξ|∞ = max
+k=1,...,d |ξk| ≤ r .= max(|r1|, |r2|).
+Further, since F ′(x) =
+1
+√
+2πe−x2/2 < 1, the function F(x) is Lipschitz with constant 1 and
+it follows from (28) that
+(ξk+1 − ξk)/ak ≥ F((ξk+1 − vk)/ak) − F((ξk − vk)/ak) ≥ δ,
+k = 1, . . . , d − 1, ak > 0.
+We find that
+ξk+1 − ξk ≥ akδ
+(this also includes the case ak = 0). We conclude tat the set E(¯ξ) ≤ c lies in the compact
+set
+K = { ¯ξ = (ξ1, . . . , ξd) ∈ Rd | |¯ξ|∞ ≤ r, ξk+1 − ξk ≥ akδ ∀k = 1, . . . , d − 1 }.
+By the continuity of E(¯ξ) the set E(¯ξ) ≤ c is a closed subset of K and therefore is
+compact.
+We take c > N .= inf E(¯ξ). Then the set E(¯ξ) ≤ c is not empty. By Proposition 1
+this set is compact and therefore the continuous function E(¯ξ) reaches the minimal value
+on it, which is evidently equal to N. We proved the existence of global minimum E(¯ξ0) =
+min E(¯ξ). The uniqueness of the minimum point is a consequence of strict convexity of the
+entropy, which is stated in Proposition 2 below. The following lemma plays a key role.
+Lemma 1. The function P(x, y) = − ln(F(x) − F(y)) is strictly convex in the half-plane
+x > y.
+7
+
+Proof. The function P(x, y) is infinitely differentiable in the domain x > y. To prove the
+lemma, we need to establish that the Hessian D2P is positive definite at every point. By
+the direct computation we find
+∂2
+∂x2P(x, y) = (F ′(x))2 − F ′′(x)(F(x) − F(y))
+(F(x) − F(y))2
+,
+∂2
+∂y2P(x, y) = (F ′(y))2 − F ′′(y)(F(y) − F(x))
+(F(x) − F(y))2
+,
+∂2
+∂x∂yP(x, y) = −
+F ′(x)F ′(y)
+(F(x) − F(y))2.
+We have to prove positive definiteness of the matrix Q = (F(x) − F(y))2D2P(x, y) with
+the components
+Q11 = (F ′(x))2 − F ′′(x)(F(x) − F(y)),
+Q22 = (F ′(y))2 − F ′′(y)(F(y) − F(x)), Q12 = Q21 = −F ′(x)F ′(y).
+Since F ′(x) = e−x2/2, then F ′′(x) = −xF ′(x) and the diagonal elements of this matrix can
+be written in the form
+Q11 = F ′(x)(x(F(x) − F(y)) + F ′(x)) =
+F ′(x)(x(F(x) − F(y)) + (F ′(x) − F ′(y))) + F ′(x)F ′(y),
+Q22 = F ′(y)(y(F(y) − F(x)) + (F ′(y) − F ′(x))) + F ′(x)F ′(y).
+By Cauchy mean value theorem there exists such a value z ∈ (y, x) that
+F ′(x) − F ′(y)
+F(x) − F(y) = F ′′(z)
+F ′(z) = −z.
+Therefore,
+Q11 = F ′(x)(F(x) − F(y))(x − z) + F ′(x)F ′(y),
+Q22 = F ′(y)(F(x) − F(y))(z − y) + F ′(x)F ′(y),
+and it follows that Q = R1 +F ′(x)F ′(y)R2, where R1 is a diagonal matrix with the positive
+diagonal elements F ′(x)(F(x)−F(y))(x−z), F ′(y)(F(x)−F(y))(z−y) while R2 =
+� 1
+−1
+−1
+1
+�
+.
+Since R1 > 0, R2 ≥ 0, then the matrix Q > 0, as was to be proved.
+Corollary 1. The functions P(x, −∞) = − ln F(x), P(+∞, x) = − ln(1 − F(x)) of single
+variable are strictly convex.
+Proof. Since 1 − F(x) = F(−x), we see that P(+∞, x) = P(−x, −∞), and it is sufficient
+to prove the strict convexity of the function P(x, −∞) = − ln F(x). By Lemma 1 in the
+limit as y → −∞ we obtain that this function is convex, moreover,
+0 ≤ (F(x))2 d2
+dx2P(x, −∞) = lim
+y→−∞ Q11 = F ′(x)(xF(x) + F ′(x)).
+If
+d2
+dx2P(x, −∞) = 0 at some point x = x0 then 0 = x0F(x0)+F ′(x0) is the minimum of the
+nonnegative function xF(x) + F ′(x). Therefore, its derivative (xF + F ′)′(x0) = 0. Since
+F ′′(x) = −xF ′(x), this derivative
+(xF + F ′)′(x0) = F(x0) + x0F ′(x0) + F ′′(x0) = F(x0) > 0.
+But this contradicts our assumption. We conclude that
+d2
+dx2P(x, −∞) > 0 and the function
+P(x, −∞) is strictly convex.
+8
+
+Proposition 2 (convexity). The entropy function E(¯ξ) is strictly convex on Ω.
+Proof. For k = 0, . . . , n − 1 we denote Pk(¯ξ) = − ln(F((ξk+1 − vk)/ak) − F((ξk − vk)/ak))
+if ak > 0, and Pk(¯ξ) = (ξk+1 − vk)2 if ak = 0. In view of (25) the entropy E(¯ξ) is a linear
+combination of the functions Pk with positive coefficients, and convexity of the entropy
+readily follows from the statements of Lemma 1 and Corollary 1. To establish the strict
+convexity, we have to demonstrate that the Hessian matrix D2E(¯ξ) is strictly positive.
+Assume that for some ζ = (ζ1, . . . , ζd) ∈ Rd
+D2E(¯ξ)ζ · ζ =
+d
+�
+i,j=1
+∂2E(¯ξ)
+∂ξi∂ξj
+ζiζj = 0.
+(31)
+Since E(¯ξ) is a linear combination of convex functions Pk(¯ξ) with positive coefficients, we
+find that
+D2Pk(¯ξ)ζ · ζ = 0
+∀k = 0, . . . , n − 1.
+This can be written in the form
+�
+i,j=k,k+1
+∂2Pk(¯ξ)
+∂ξi∂ξj
+ζiζj = 0 if 0 < k < n − 1, ak > 0;
+∂2Pk(¯ξ)
+∂ξ2
+k+1
+ζ2
+k+1 if k = 0 or ak = 0.
+In view of Lemma 1 and Corollary 1 the functions Pk in above equalities are strictly convex
+as functions of either two variables (ξk, ξk+1) or single variable ξk+1.
+Therefore, these
+equalities imply that in any case ζk+1 = 0, k = 0, . . . , n − 2, and ζn = 0 if an−1 = 0 (when
+d = n). We conclude that all coordinates ζi = 0, i = 1, . . . , d. Hence, equality (31) can
+hold only for ζ = 0 and the matrix D2P(¯ξ) > 0 for all ¯ξ ∈ Ω. This completes the proof.
+4
+The variational formulation
+Let ¯ξ0 = (ξ1, . . . , ξd) ∈ Ω be the unique minimum point of E(¯ξ).
+The necessary and
+sufficient condition for ¯ξ0 to be a minimum point is the following one
+∇E(¯ξ0) · p ≥ 0
+∀p ∈ T(¯ξ0) = { p ∈ Rd | ∃h > 0 ¯ξ0 + hp ∈ Ω },
+(32)
+so that T(¯ξ0) is the tangent cone to Ω at the point ¯ξ0. If ¯ξ0 ∈ Int Ω then T(¯ξ0) = Rd
+and (32) reduces to the requirement ∇E(¯ξ0) = 0. As we have already demonstrated, this
+requirement coincides with jump conditions (17), (18), (19), (20) for all k = 1, . . . , d. But
+these conditions are equivalent to the statement that the function (13) is an e.s. of (1),
+(3). In the general situation when ¯ξ0 can belong to the boundary of Ω, the coordinates of
+¯ξ0 may coincides. Let ξk = · · · = ξl = c be a maximal family of coinciding coordinates,
+that is, ξk−1 < ξk = ξl < ξl+1 (it is possible here that k = l). Then, as is easy to realize,
+the vector p = (p1, . . . , pd), with arbitrary increasing coordinates pk ≤ · · · ≤ pl and with
+zero remaining coordinates, belong to the tangent cone T(¯ξ0). In view of (32)
+l
+�
+i=k
+∂
+∂ξi
+E(¯ξ0)pi ≥ 0
+9
+
+for any such a vector. Using the summation by parts formula, we realize that the above
+condition is equivalent to the following requirements
+l
+�
+i=k
+∂
+∂ξi
+E(¯ξ0) = 0,
+(33)
+l
+�
+i=j
+∂
+∂ξi
+E(¯ξ0) ≥ 0, k < j ≤ l.
+(34)
+Recall that ai = 0 for k ≤ i < l. By the direct computation we find
+∂
+∂ξi
+E(¯ξ0) = (ui − ui−1)(ξi − vi−1),
+k < i < l,
+∂
+∂ξk
+E(¯ξ0) =
+� (uk − uk−1)(ξk − vk−1)
+,
+ak−1 = 0,
+−A(u)′(c−)
+,
+ak−1 > 0;
+∂
+∂ξl
+E(¯ξ0) = A(u)′(c+),
+where A(u)′(c±) are given by (22), (23). Putting these expressions into (33), (34), we
+obtain exactly the jump conditions (21), (24). Therefore, the function (13) corresponding
+to the point ¯ξ0 is an e.s. of (1), (3). Conversely, if (13) is an e.s. then relations (33), (34)
+holds for all groups of coinciding coordinates. As is easy to verify, this is equivalent to the
+criterion (32). We have proved our main result.
+Theorem 1. The function (13) is an e.s. of (1), (3) if and only if ¯ξ0 = (ξ1, . . . , ξd) is the
+minimum point of the entropy E(¯ξ).
+Remark 1. Adding to the entropy (25) the constant
+�
+k=0,...,n−1,ak>0
+(ak)2(uk+1 − uk) ln((uk+1 − uk)/ak),
+we obtain the alternative variant of the entropy
+E1(¯ξ) = −
+�
+k=0,...,n−1,ak>0
+(ak)2(uk+1 − uk) ln
+�F((ξk+1 − vk)/ak) − F((ξk − vk)/ak)
+(uk+1 − uk)/ak
+�
++1
+2
+�
+k=0,...,n−1,ak=0
+(uk+1 − uk)(ξk+1 − vk)2. (35)
+If we consider the values vk, ak as a piecewise constant approximation of an arbitrary
+velocity function v(u) and, respectively, a diffusion function a(u) ≥ 0 then, passing in
+(35) to the limit as max(uk+1 − uk) → 0, we find that the entropy E1(¯ξ) turns into the
+variational functional
+J(ξ) = −
+�
+{u∈[α,β],a(u)>0}
+(a(u))2 ln(F ′((ξ(u) − v(u))/a(u))ξ′(u))du+
+1
+2
+�
+{u∈[α,β],a(u)=0}
+(ξ(u) − v(u))2du,
+10
+
+where ξ(u) is an increasing function on [α, β], which is expected to be the inverse function
+to a self-similar solution u = u(ξ) of the problem (1), (3). Taking into account that
+ln(F ′((ξ(u) − v(u))/a(u))ξ′(u)) = ln F ′((ξ(u) − v(u))/a(u)) + ln ξ′(u) =
+−(ξ(u) − v(u))2
+2a2(u)
++ ln ξ′(u),
+we may simplify the expression for the functional J(ξ)
+J(ξ) =
+� β
+α
+[(ξ(u) − v(u))2/2 − (a(u))2 ln(ξ′(u))]du.
+(36)
+We see that this functional is strictly convex. The corresponding Euler-Lagrange equation
+has the form
+ξ(u) − v(u) + ((a(u))2/ξ′(u))′ = 0.
+(37)
+Since u′(ξ) = 1/ξ′(u), u = u(ξ), we can transform (37) as follows
+ξ(u) − v(u) + ((a(u))2u′(ξ))′
+u = 0.
+Multiplying this equation by u′(ξ), we obtain the equation
+(a2u′)′ = (v − ξ)u′,
+u = u(ξ),
+which is exactly our equation (1) written in the self-similar variable.
+Remark 2. In the case of conservation laws (2) the e.s. u = u(ξ) of (2), (3) is piecewise
+constant, and, by expression (13),
+u(ξ) = uk,
+ξk < ξ < ξk+1, k = 0, . . . , n,
+where −∞ = ξ0 < ξ1 ≤ · · · ≤ ξn < ξn+1 = +∞. In this case the entropy function is
+particularly simple, it is the quadratic function
+E(¯ξ) = 1
+2
+n
+�
+k=1
+(uk − uk−1)(ξk − vk−1)2,
+defined on the closed polyhedral cone
+Ω = { ¯ξ = (ξ1, . . . , ξn) ∈ Rn | ξk+1 ≥ ξk ∀k = 1, . . . , n − 1 }.
+Existence and uniqueness of a minimal point in this case is trivial. By Theorem 1 and
+Remark 1 we obtain new, variational formulation of the entropy solution.
+Acknowledgments
+The research was supported by the Russian Science Foundation, grant 22-21-00344.
+11
+
+References
+[1] J. Carrillo, Entropy solutions for nonlinear degenerate problems, Arch. Ration. Mech.
+Anal., 147 (1999), 269–361.
+[2] S. N. Kruzhkov, First order quasilinear equations in several independent variables, Mat.
+Sb. (N.S.), 81 (1970), 228–255.
+[3] O. A. Oleinik, Uniqueness and stability of the generalized solution of the Cauchy prob-
+lem for a quasi-linear equation, Uspekhi Mat. Nauk, 14:2(86) (1959), 165–170.
+12
+

KdFQT4oBgHgl3EQfTTZa/content/tmp_files/load_file.txt

@@ -0,0 +1,335 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf,len=334
+page_content='On the structure of entropy solutions to the Riemann  problem for a degenerate nonlinear parabolic equation  Evgeny Yu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' Panov  Yaroslav-the-Wise Novgorod State University, Veliky Novgorod, Russian Federation,  Research and Development Center, Veliky Novgorod, Russian Federation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content='  Abstract  We find an explicit form of entropy solutions to a Riemann problem for a de-  generate nonlinear parabolic equation with piecewise constant velocity and diffusion  coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' It is demonstrated that this solution corresponds to the minimum point  of some strictly convex function of a finite number of variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content='  1  Introduction  In a half-plane Π = {(t, x) | t > 0, x ∈ R}, we consider a nonlinear parabolic equation  ut + v(u)ux − t(a2(u)ux)x = 0,  (1)  where v(u), a(u) ∈ L∞(R), a(u) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' Since the diffusion coefficient a(u) may take zero  value, equation (1) is degenerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' In the case when a(u) ≡ 0 it reduces to a first order  conservation law  ut + ϕ(u)x = 0,  (2)  where ϕ′(u) = v(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' Similarly, a general equation (1) can be written in the conservative  form  ut + ϕ(u)x − tA(u)xx = 0  with A′(u) = a2(u), which allows to define weak solutions of this equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' Unfortunately,  weak solutions to a Cauchy problem for equation (1) are not unique in general, and some  additional entropy conditions are required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' We consider the Cauchy problem with initial  data  u(0, x) = u0(x),  (3)  where u0(x) ∈ L∞(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' Recall the notion of entropy solution (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' for short) in the sense of  Carrillo [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content='  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' A function u = u(t, x) ∈ L∞(Π) is called an e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' of (1), (3) if  (i) the distribution A(u) ∈ L2  loc(Π);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content='  (ii) for all k ∈ R  |u − k|t + (sign(u − k)(ϕ(u) − ϕ(k)))x − (t sign(u − k)(A(u) − A(k)))xx ≤ 0  (4)  in the sense of distributions (in D′(Π));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content='  (iii) ess lim  t→0  u(t, ·) = u0 in L1  loc(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content='  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content='13292v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content='AP]  30 Jan 2023  Entropy condition (4) means that for each nonnegative test function f = f(t, x) ∈  C2  0(Π)  �  Π  [|u − k|ft + sign(u − k)((ϕ(u) − ϕ(k))fx + t(A(u) − A(k))fxx)]dtdx ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content='  (5)  In the case of conservation laws (2) the notion of e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' reduces to the notion of generalized  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' in the sense of Kruzhkov [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' Taking in (4) k = ±M, M ≥ ∥u∥∞, we derive that  ut + ϕ(u)x − tA(u)xx = 0 in D′(Π),  that is, an e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' u of (1), (3) is a weak solution of this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' It is known that an e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' of (1),  (3) always exists and is unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' In general multidimensional setting this was demonstrated  in [2] for conservation laws and in [1] for the general case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' If to be precise, in [1] the case of  usual diffusion term A(u)xx was studied but the proofs can be readily adapted to the case  of the self-similar diffusion tA(u)xx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content='  If u = u(t, x) is a piecewise C2-smooth e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' of equation (1) then it must satisfy this  equation in classic sense in each smoothness domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content='  Applying relation (1) to a test  function f = f(t, x) ∈ C2  0(Π) supported in a neighborhood of a discontinuity line x = x(t)  and integrating by parts, we then obtain the identity  (−x′(t)[u] + [ϕ(u)] − t[A(u)x])f + t[A(u)]fx = 0  (6)  a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' on the line x = x(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' Here we denote by [w] the jump of a function w = w(t, x) on the  line x = x(t) so that  [w] = w(t, x(t)+) − w(t, x(t)−),  where w(t, x(t)±) =  lim  y→x(t)± w(t, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content='  Since the functions f, fx are arbitrary and independent on the line x = x(t), identity (6)  implies the following two relations of Rankine-Hugoniot type  [A(u)] = 0,  (7)  −x′(t)[u] + [ϕ(u)] − t[A(u)x] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content='  (8)  Similarly, it follows from entropy relation (5), after integration by parts, that  (−x′(t)[|u − k|] + [sign(u − k)(ϕ(u) − ϕ(k))] − t[sign(u − k)A(u)x])f+  t[sign(u − k)(A(u) − A(k))]fx ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content='  (9)  Since the function A(u) increases, it follows from (7) that A(u) = const when u lies between  the values u(t, x(t)−) and u(t, x(t)+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' This implies that [sign(u − k)(A(u) − A(k))] = 0  and in view of arbitrariness of f ≥ 0 it follows from (9) that  − x′(t)[|u − k|] + [sign(u − k)(ϕ(u) − ϕ(k))] − t[sign(u − k)A(u)x] ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content='  (10)  In the case when k lies out of the interval with the endpoints u± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content='= u(t, x(t)±) relation (10)  follows from (8) and fulfils with equality sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' When u− < k < u+ this relation reads  −x′(t)(u+ + u− − 2k) + ϕ(u+) + ϕ(u−) − 2ϕ(k) − t(A(u)+  x + A(u)−  x ) ≤ 0,  2  where A(u)±  x = A(u)x(t, x(t)±).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' Adding (8) to this relation and dividing the result by 2,  we arrive at the following analogue of the famous Oleinik condition (see [3]) known for  conservation laws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content='  − x′(t)(u+ − k) + ϕ(u+) − tA(u)+  x − ϕ(k) ≤ 0  ∀k ∈ [u−, u+].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content='  (11)  In the case u+ < u− this condition has the form  − x′(t)(u+ − k) + ϕ(u+) − tA(u)+  x − ϕ(k) ≥ 0  ∀k ∈ [u+, u−]  (12)  and can be derived similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content='  Geometric interpretation of these conditions is that the  graph of the flux function ϕ(u) lies not below (not above) of the segment connecting the  points (u−, ϕ(u−) − tA(u)−  x ), (u+, ϕ(u+) − tA(u)+  x ) when u− ≤ u ≤ u+ (respectively, when  u+ ≤ u ≤ u−), see Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' We take here into account that in view of condition (8)  the vector (−x′(t), 1) is a normal to the indicated segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' We also notice that it follows  from relations (11), (12) with k = u± and from the Rankine-Hugoniot condition (8) that  A(u)±  x ≥ 0 (A(u)±  x ≤ 0) whenever u+ > u− (u+ < u−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content="  u  u-  u+  y=φ(u)  y  tA(u)x tA(u)x  +  (-x',1)  k  Figure 1: Oleinik condition." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content='  2  The case of piecewise constant coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content='  Below we will assume that the functions v(u), a(u) are piecewise constant, v(u) = vk,  a(u) = ak when uk < u < uk+1, k = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' , n − 1, where  α = u0 < u1 < · · · < un−1 < un = β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content='  We will study problem (1), (3) with the Riemann data u0(x) =  � α,  x < 0,  β,  x > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content='  Since this  problem is invariant under the scaling transformations t → λt, x → λx, λ > 0 then, by  the uniqueness, the e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' u = u(t, x) is self-similar: u(t, x) = u(λt, λx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' This implies that  u = u(x/t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' Suppose that ak > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' Then in a domain where uk < u(ξ) < uk+1 with ξ = x/t  equation (1) reduces to the second order ODE  (vk − ξ)u′ − a2  ku′′ = 0,  3  the general solution of which is u = C1F((ξ − vk)/ak) + C2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' C1, C2 = const, where  F(z) =  1  √  2π  � z  −∞  e−s2/2ds  is the error function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' Therefore, it is natural to seek the e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' of our problem in the following  form  u(ξ) =  � uk +  uk+1−uk  F((ξk+1−vk)/ak)−F((ξk−vk)/ak)(F((ξ − vk)/ak) − F((ξk − vk)/ak))  ,  ak > 0,  uk  ,  ak = 0,  (13)  ξk < ξ < ξk+1, k = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' , d,  where  d =  � n − 1  ,  an−1 > 0,  n  ,  an−1 = 0,  − ∞ = ξ0 < ξ1 ≤ · · · ≤ ξd < ξd+1 = +∞,  and we agree that an = 0, F(−∞) = 0, F(+∞) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' We also assume that ξk+1 > ξk  whenever ak > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' The rays x = ξkt for finite ξk are (weak or strong) discontinuity lines of  u, they correspond to discontinuity points ξk of the function u(ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' Observe that conditions  (7), (8) turns into the following relations at points ξk  [A(u)] = A(u(ξk+)) − A(u(ξk−)) = 0,  (14)  −ξk[u] + [ϕ(u)] − [A(u)′] = −ξk(u(ξk+) − u(ξk−)) + ϕ(u(ξk+))−  ϕ(u(ξk−)) − A(u)′(ξk+) + A(u)′(ξk−) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content='  (15)  Here w(ξk±) denotes unilateral limits of a function w(ξ) at the point ξk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' Similarly, the  Oleinik condition (11) reads  − ξk(u(ξk+) − k) + ϕ(u(ξk+)) − A(u)′(ξk+) − ϕ(k) ≤ 0  ∀k ∈ [u(ξk−), u(ξk+)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content='  (16)  Notice that our solution (13) is a nonstrictly increasing function of the self-similar variable  ξ and, therefore, u(ξk−) ≤ u(ξk+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content='  Let us firstly analyze the solution (13) in the case ξk−1 < ξk < ξk+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' If ak−1, ak > 0  then u(ξk−) = u(ξk+) = uk so that condition (14) fulfils while (15) reduces to the equality  [A(u)′] = 0, which is revealed as  ak(uk+1 − uk)  F((ξk+1 − vk)/ak) − F((ξk − vk)/ak)F ′((ξk − vk)/ak) =  ak−1(uk − uk−1)  F((ξk − vk−1)/ak−1) − F((ξk−1 − vk−1)/ak−1)F ′((ξk − vk−1)/ak−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content='  (17)  In this situation ξ = ξk is a weak discontinuity point, the function u(ξ) itself is continuous,  only its derivative u′(ξ) may be discontinuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' Moreover, it follows from (17) that both  functions u(ξ) and u′(ξ) are continuous at point ξk if ak = ak−1 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content='  If ak−1 > ak = 0 then again u(ξ) is continuous at uk and (15) reduces to the relation  ak−1(uk − uk−1)  F((ξk − vk−1)/ak−1) − F((ξk−1 − vk−1)/ak−1)F ′((ξk − vk−1)/ak−1) = 0,  (18)  4  which is impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' If ak > ak−1 = 0 then u(ξk−) = uk−1 < uk = u(ξk+), that is, ξk is  a strong discontinuity point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' Condition (14) holds because A(u) is constant on [uk−1, uk]  (A′(u) = a2  k−1 = 0) while (15) turns into  (vk−1 − ξk)(uk − uk−1) −  ak(uk+1 − uk)  F((ξk+1 − vk)/ak) − F((ξk − vk)/ak)F ′((ξk − vk)/ak) = 0, (19)  where we use the fact that ϕ(uk) − ϕ(uk−1) = vk−1(uk − uk−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' It remains to analyze the  situation when ak = ak−1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' In this case again u(ξk−) = uk−1 < uk = u(ξk+) and  A(uk−1) = A(uk) while condition (15) turns into the simple relation  ξk = vk−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content='  (20)  Finally, since the function ϕ(u) is affine on the segment [uk−1, uk] and A(u)′(ξk±) ≥ 0, then  entropy relation (16) is always satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content='  Now we consider the case when there exists a nontrivial family of mutually equaled  values ξi, ξi = ξk for i = k, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' , l, where l > k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' We can assume that this family is maximal,  that is,  −∞ ≤ ξk−1 < ξk = · · · = ξl < ξl+1 ≤ +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content='  Then ai = 0 for i = k, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' , l − 1, and the point ξ = c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content='= ξk is a discontinuity point of u(ξ)  with the unilateral limits  u(c+) = ul, u(c−) = uk′,  where k′ =  � k  ,  ak−1 > 0,  k − 1  ,  ak−1 = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content='  Since a(u) = 0 for u(c−) < u < u(c+), we find that A(u(c−)) = A(u(c+)) and condition  (14) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' Further, we notice that  l−1  �  i=k′  (−ξi+1(ui+1 − ui)) = −c  l−1  �  i=k′  (ui+1 − ui) = −c(ul − uk′),  l−1  �  i=k′  vi(ui+1 − ui) =  l−1  �  i=k′  (ϕ(ui+1) − ϕ(ui)) = ϕ(ul) − ϕ(uk′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content='  Therefore, condition (15) can be written in the form  l−1  �  i=k′  (vi − ξi+1)(ui+1 − ui) − (A(u)′(c+) − A(u)′(c−)) = 0,  (21)  where, as is easy to verify,  A(u)′(c−) =  �  0  ,  ak−1 = 0,  ak−1(uk−uk−1)  F((ξk−vk−1)/ak−1)−F((ξk−1−vk−1)/ak−1)F ′((ξk − vk−1)/ak−1)  ,  ak−1 > 0,  (22)  A(u)′(c+) =  �  0  ,  al = 0,  al(ul+1−ul)  F((ξl+1−vl)/al)−F((ξl−vl)/al)F ′((ξl − vl)/al)  ,  al > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content='  (23)  5  In the similar way we can write the Oleinik condition (16) as follows  l−1  �  i=j  (vi − ξi+1)(ui+1 − ui) − A(u)′(c+) ≤ 0  k′ < j < l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content='  (24)  We use here the fact the function ϕ(u) is piecewise affine and, therefore, it is enough to  verify the Oleinik condition (16) only at the nodal points k = uj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content='  The above reasoning remains valid also in the case when l = k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' In this case, relation  (21) reduces to one of conditions (17), (18), (19), (20) while (24) is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content='  3  The entropy function  We introduce the convex cone Ω ⊂ Rd consisting of points ¯ξ = (ξ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' , ξd) with increasing  coordinates, ξ1 ≤ ξ2 ≤ · · · ≤ ξd such that ξk+1 > ξk whenever ak > 0, k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' , d − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content='  Each point ¯ξ ∈ Ω determines a function u(ξ) in correspondence with formula (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' Assume  firstly that ¯ξ ∈ Int Ω, that is, the values ξk are strictly increasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' Then conditions (17),  (18), (19), (20) coincides with the equality  ∂  ∂ξk E(¯ξ) = 0, where  E(¯ξ) = −  �  k=0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=',n−1,ak>0  (ak)2(uk+1 − uk) ln(F((ξk+1 − vk)/ak) − F((ξk − vk)/ak))+  1  2  �  k=0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=',n−1,ak=0  (uk+1 − uk)(ξk+1 − vk)2,  ¯ξ = (ξ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' , ξd) ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content='  (25)  We will call this function the entropy because it depends only on the discontinuities of a  solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' Thus, for ¯ξ ∈ Int Ω the e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' (13) corresponds to a critical point of the entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' We  are going to demonstrate that the entropy is strictly convex and coercive in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' Therefore,  it has a unique global minimum point in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' In the case when this minimum point lies in  Int Ω it is a unique critical point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content='  Obviously, E(¯ξ) ∈ C∞(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' Notice that for all k = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' , n − 1, such that ak > 0  ln(F((ξk+1 − vk)/ak) − F((ξk − vk)/ak)) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content='  Therefore, all terms in expression (25) are nonnegative and, in particular, E(¯ξ) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content='  Proposition 1 (coercivity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' The sets E(¯ξ) ≤ c are compact for each constant c ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' If E(¯ξ) ≤ c then it follows from nonnegativity of all terms in (25) that for all  k = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' , n − 1  −(ak)2(uk+1 − uk) ln(F((ξk+1 − vk)/ak) − F((ξk − vk)/ak)) ≤ E(¯ξ) ≤ c if ak > 0,  (26)  (uk+1 − uk)(ξk+1 − vk)2/2 ≤ c if ak = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content='  (27)  Relation (26) implies the estimate  F((ξk+1 − vk)/ak) − F((ξk − vk)/ak) ≥ δ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content='= exp(−c/m) > 0,  (28)  where m =  min  k=0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=',n−1,ak>0(ak)2(uk+1 − uk) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' If a0 > 0 relation (28) with k = 0 reads  F((ξ1 − v0)/a0) > δ (notice that F((ξ0 − v0)/a0) = F(−∞) = 0), which implies that  ξ1 ≥ v0 + a0F −1(δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content='  6  On the other hand, if a0 = 0 then (u1 − u0)(ξ1 − v0)2 ≤ 2c, in view of (27) with k = 0, and  ξ1 ≥ v0 − (2c/(u1 − u0))1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content='  In any case,  ξ1 ≥ r1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content='= v0 + min(a0F −1(δ), −(2c/(u1 − u0))1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content='  (29)  To get an upper bound, we remark that in the case an−1 > 0 it follows from (28) with  k = d = n − 1 that F(−(ξn−1 − vn−1)/an−1) = 1 − F((ξn−1 − vn−1)/an−1) ≥ δ (observe that  F((ξn − vn−1)/an−1) = F(+∞) = 1), which implies the estimate  ξd ≤ vn−1 − an−1F −1(δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content='  If an−1 = 0 then d = n and in view of inequality (27) with k = n − 1 we find (un −  un−1)(ξn − vn−1)2/2 ≤ c, that is,  ξd ≤ vn−1 + (2c/(un − un−1))1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content='  In both cases  ξd ≤ r2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content='= vn−1 + max(−an−1F −1(δ), (2c/(un − un−1))1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content='  (30)  Since all coordinates of ¯ξ lie between ξ1 and ξd, estimates (29), (30) imply the bound  |¯ξ|∞ = max  k=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=',d |ξk| ≤ r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content='= max(|r1|, |r2|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content='  Further, since F ′(x) =  1  √  2πe−x2/2 < 1, the function F(x) is Lipschitz with constant 1 and  it follows from (28) that  (ξk+1 − ξk)/ak ≥ F((ξk+1 − vk)/ak) − F((ξk − vk)/ak) ≥ δ,  k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' , d − 1, ak > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content='  We find that  ξk+1 − ξk ≥ akδ  (this also includes the case ak = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' We conclude tat the set E(¯ξ) ≤ c lies in the compact  set  K = { ¯ξ = (ξ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' , ξd) ∈ Rd | |¯ξ|∞ ≤ r, ξk+1 − ξk ≥ akδ ∀k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' , d − 1 }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content='  By the continuity of E(¯ξ) the set E(¯ξ) ≤ c is a closed subset of K and therefore is  compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content='  We take c > N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content='= inf E(¯ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' Then the set E(¯ξ) ≤ c is not empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' By Proposition 1  this set is compact and therefore the continuous function E(¯ξ) reaches the minimal value  on it, which is evidently equal to N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' We proved the existence of global minimum E(¯ξ0) =  min E(¯ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' The uniqueness of the minimum point is a consequence of strict convexity of the  entropy, which is stated in Proposition 2 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' The following lemma plays a key role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content='  Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' The function P(x, y) = − ln(F(x) − F(y)) is strictly convex in the half-plane  x > y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content='  7  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' The function P(x, y) is infinitely differentiable in the domain x > y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' To prove the  lemma, we need to establish that the Hessian D2P is positive definite at every point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' By  the direct computation we find  ∂2  ∂x2P(x, y) = (F ′(x))2 − F ′′(x)(F(x) − F(y))  (F(x) − F(y))2  ,  ∂2  ∂y2P(x, y) = (F ′(y))2 − F ′′(y)(F(y) − F(x))  (F(x) − F(y))2  ,  ∂2  ∂x∂yP(x, y) = −  F ′(x)F ′(y)  (F(x) − F(y))2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content='  We have to prove positive definiteness of the matrix Q = (F(x) − F(y))2D2P(x, y) with  the components  Q11 = (F ′(x))2 − F ′′(x)(F(x) − F(y)),  Q22 = (F ′(y))2 − F ′′(y)(F(y) − F(x)), Q12 = Q21 = −F ′(x)F ′(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content='  Since F ′(x) = e−x2/2, then F ′′(x) = −xF ′(x) and the diagonal elements of this matrix can  be written in the form  Q11 = F ′(x)(x(F(x) − F(y)) + F ′(x)) =  F ′(x)(x(F(x) − F(y)) + (F ′(x) − F ′(y))) + F ′(x)F ′(y),  Q22 = F ′(y)(y(F(y) − F(x)) + (F ′(y) − F ′(x))) + F ′(x)F ′(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content='  By Cauchy mean value theorem there exists such a value z ∈ (y, x) that  F ′(x) − F ′(y)  F(x) − F(y) = F ′′(z)  F ′(z) = −z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content='  Therefore,  Q11 = F ′(x)(F(x) − F(y))(x − z) + F ′(x)F ′(y),  Q22 = F ′(y)(F(x) − F(y))(z − y) + F ′(x)F ′(y),  and it follows that Q = R1 +F ′(x)F ′(y)R2, where R1 is a diagonal matrix with the positive  diagonal elements F ′(x)(F(x)−F(y))(x−z), F ′(y)(F(x)−F(y))(z−y) while R2 =  � 1  −1  −1  1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content='  Since R1 > 0, R2 ≥ 0, then the matrix Q > 0, as was to be proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content='  Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' The functions P(x, −∞) = − ln F(x), P(+∞, x) = − ln(1 − F(x)) of single  variable are strictly convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' Since 1 − F(x) = F(−x), we see that P(+∞, x) = P(−x, −∞), and it is sufficient  to prove the strict convexity of the function P(x, −∞) = − ln F(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' By Lemma 1 in the  limit as y → −∞ we obtain that this function is convex, moreover,  0 ≤ (F(x))2 d2  dx2P(x, −∞) = lim  y→−∞ Q11 = F ′(x)(xF(x) + F ′(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content='  If  d2  dx2P(x, −∞) = 0 at some point x = x0 then 0 = x0F(x0)+F ′(x0) is the minimum of the  nonnegative function xF(x) + F ′(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' Therefore, its derivative (xF + F ′)′(x0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' Since  F ′′(x) = −xF ′(x), this derivative  (xF + F ′)′(x0) = F(x0) + x0F ′(x0) + F ′′(x0) = F(x0) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content='  But this contradicts our assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' We conclude that  d2  dx2P(x, −∞) > 0 and the function  P(x, −∞) is strictly convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content='  8  Proposition 2 (convexity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' The entropy function E(¯ξ) is strictly convex on Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' For k = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' , n − 1 we denote Pk(¯ξ) = − ln(F((ξk+1 − vk)/ak) − F((ξk − vk)/ak))  if ak > 0, and Pk(¯ξ) = (ξk+1 − vk)2 if ak = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' In view of (25) the entropy E(¯ξ) is a linear  combination of the functions Pk with positive coefficients, and convexity of the entropy  readily follows from the statements of Lemma 1 and Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' To establish the strict  convexity, we have to demonstrate that the Hessian matrix D2E(¯ξ) is strictly positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content='  Assume that for some ζ = (ζ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' , ζd) ∈ Rd  D2E(¯ξ)ζ · ζ =  d  �  i,j=1  ∂2E(¯ξ)  ∂ξi∂ξj  ζiζj = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content='  (31)  Since E(¯ξ) is a linear combination of convex functions Pk(¯ξ) with positive coefficients, we  find that  D2Pk(¯ξ)ζ · ζ = 0  ∀k = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' , n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content='  This can be written in the form  �  i,j=k,k+1  ∂2Pk(¯ξ)  ∂ξi∂ξj  ζiζj = 0 if 0 < k < n − 1, ak > 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content='  ∂2Pk(¯ξ)  ∂ξ2  k+1  ζ2  k+1 if k = 0 or ak = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content='  In view of Lemma 1 and Corollary 1 the functions Pk in above equalities are strictly convex  as functions of either two variables (ξk, ξk+1) or single variable ξk+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content='  Therefore, these  equalities imply that in any case ζk+1 = 0, k = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' , n − 2, and ζn = 0 if an−1 = 0 (when  d = n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' We conclude that all coordinates ζi = 0, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' , d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' Hence, equality (31) can  hold only for ζ = 0 and the matrix D2P(¯ξ) > 0 for all ¯ξ ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content='  4  The variational formulation  Let ¯ξ0 = (ξ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' , ξd) ∈ Ω be the unique minimum point of E(¯ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content='  The necessary and  sufficient condition for ¯ξ0 to be a minimum point is the following one  ∇E(¯ξ0) · p ≥ 0  ∀p ∈ T(¯ξ0) = { p ∈ Rd | ∃h > 0 ¯ξ0 + hp ∈ Ω },  (32)  so that T(¯ξ0) is the tangent cone to Ω at the point ¯ξ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' If ¯ξ0 ∈ Int Ω then T(¯ξ0) = Rd  and (32) reduces to the requirement ∇E(¯ξ0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' As we have already demonstrated, this  requirement coincides with jump conditions (17), (18), (19), (20) for all k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' , d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' But  these conditions are equivalent to the statement that the function (13) is an e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' of (1),  (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' In the general situation when ¯ξ0 can belong to the boundary of Ω, the coordinates of  ¯ξ0 may coincides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' Let ξk = · · · = ξl = c be a maximal family of coinciding coordinates,  that is, ξk−1 < ξk = ξl < ξl+1 (it is possible here that k = l).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' Then, as is easy to realize,  the vector p = (p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' , pd), with arbitrary increasing coordinates pk ≤ · · · ≤ pl and with  zero remaining coordinates, belong to the tangent cone T(¯ξ0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' In view of (32)  l  �  i=k  ∂  ∂ξi  E(¯ξ0)pi ≥ 0  9  for any such a vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' Using the summation by parts formula, we realize that the above  condition is equivalent to the following requirements  l  �  i=k  ∂  ∂ξi  E(¯ξ0) = 0,  (33)  l  �  i=j  ∂  ∂ξi  E(¯ξ0) ≥ 0, k < j ≤ l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content='  (34)  Recall that ai = 0 for k ≤ i < l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' By the direct computation we find  ∂  ∂ξi  E(¯ξ0) = (ui − ui−1)(ξi − vi−1),  k < i < l,  ∂  ∂ξk  E(¯ξ0) =  � (uk − uk−1)(ξk − vk−1)  ,  ak−1 = 0,  −A(u)′(c−)  ,  ak−1 > 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content='  ∂  ∂ξl  E(¯ξ0) = A(u)′(c+),  where A(u)′(c±) are given by (22), (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' Putting these expressions into (33), (34), we  obtain exactly the jump conditions (21), (24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' Therefore, the function (13) corresponding  to the point ¯ξ0 is an e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' of (1), (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' Conversely, if (13) is an e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' then relations (33), (34)  holds for all groups of coinciding coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' As is easy to verify, this is equivalent to the  criterion (32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' We have proved our main result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' The function (13) is an e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' of (1), (3) if and only if ¯ξ0 = (ξ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' , ξd) is the  minimum point of the entropy E(¯ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content='  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' Adding to the entropy (25) the constant  �  k=0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=',n−1,ak>0  (ak)2(uk+1 − uk) ln((uk+1 − uk)/ak),  we obtain the alternative variant of the entropy  E1(¯ξ) = −  �  k=0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=',n−1,ak>0  (ak)2(uk+1 − uk) ln  �F((ξk+1 − vk)/ak) − F((ξk − vk)/ak)  (uk+1 − uk)/ak  �  +1  2  �  k=0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=',n−1,ak=0  (uk+1 − uk)(ξk+1 − vk)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' (35)  If we consider the values vk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' ak as a piecewise constant approximation of an arbitrary  velocity function v(u) and,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' respectively,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' a diffusion function a(u) ≥ 0 then,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' passing in  (35) to the limit as max(uk+1 − uk) → 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' we find that the entropy E1(¯ξ) turns into the  variational functional  J(ξ) = −  �  {u∈[α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content='β],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content='a(u)>0}  (a(u))2 ln(F ′((ξ(u) − v(u))/a(u))ξ′(u))du+  1  2  �  {u∈[α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content='β],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content='a(u)=0}  (ξ(u) − v(u))2du,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content='  10  where ξ(u) is an increasing function on [α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' β],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' which is expected to be the inverse function  to a self-similar solution u = u(ξ) of the problem (1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' Taking into account that  ln(F ′((ξ(u) − v(u))/a(u))ξ′(u)) = ln F ′((ξ(u) − v(u))/a(u)) + ln ξ′(u) =  −(ξ(u) − v(u))2  2a2(u)  + ln ξ′(u),  we may simplify the expression for the functional J(ξ)  J(ξ) =  � β  α  [(ξ(u) − v(u))2/2 − (a(u))2 ln(ξ′(u))]du.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content='  (36)  We see that this functional is strictly convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' The corresponding Euler-Lagrange equation  has the form  ξ(u) − v(u) + ((a(u))2/ξ′(u))′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content='  (37)  Since u′(ξ) = 1/ξ′(u), u = u(ξ), we can transform (37) as follows  ξ(u) − v(u) + ((a(u))2u′(ξ))′  u = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content='  Multiplying this equation by u′(ξ), we obtain the equation  (a2u′)′ = (v − ξ)u′,  u = u(ξ),  which is exactly our equation (1) written in the self-similar variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' In the case of conservation laws (2) the e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' u = u(ξ) of (2), (3) is piecewise  constant, and, by expression (13),  u(ξ) = uk,  ξk < ξ < ξk+1, k = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' , n,  where −∞ = ξ0 < ξ1 ≤ · · · ≤ ξn < ξn+1 = +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' In this case the entropy function is  particularly simple, it is the quadratic function  E(¯ξ) = 1  2  n  �  k=1  (uk − uk−1)(ξk − vk−1)2,  defined on the closed polyhedral cone  Ω = { ¯ξ = (ξ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' , ξn) ∈ Rn | ξk+1 ≥ ξk ∀k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' , n − 1 }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content='  Existence and uniqueness of a minimal point in this case is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' By Theorem 1 and  Remark 1 we obtain new, variational formulation of the entropy solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content='  Acknowledgments  The research was supported by the Russian Science Foundation, grant 22-21-00344.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content='  11  References  [1] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' Carrillo, Entropy solutions for nonlinear degenerate problems, Arch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' Ration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content='  Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=', 147 (1999), 269–361.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content='  [2] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' Kruzhkov, First order quasilinear equations in several independent variables, Mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content='  Sb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' (N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' ), 81 (1970), 228–255.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content='  [3] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' Oleinik, Uniqueness and stability of the generalized solution of the Cauchy prob-  lem for a quasi-linear equation, Uspekhi Mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content=' Nauk, 14:2(86) (1959), 165–170.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}
+page_content='  12' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFQT4oBgHgl3EQfTTZa/content/2301.13292v1.pdf'}

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+Status of leptoquark models after LHC Run-2 and discovery
+prospects at future colliders
+Nishita Desai∗
+Department of Theoretical Physics,
+Tata Institute of Fundamental Research,
+Mumbai, India 400005
+Amartya Sengupta
+Meghnad Saha Pally, Burdwan, India, 713104 †
+We study limits from dilepton searches on leptoquark completions to the Standard
+Model in the parameter space motivated by anomalies in the b → s sector. After a full
+Run-2 analysis by LHCb, the disparity in lepton flavour violation has disappeared.
+However, the mismatch in angular distributions as well as in Bs → µ+µ− partial
+width is still unresolved and still implies a possible new physics contribution. We
+probe three models of leptoquarks — scalar models S3 and R2 as well as vector
+leptoquark model U1 using non-resonant dilepton searches to place limit on both the
+mass and couplings to SM fermions. The exclusions of leptoquarks coupling either
+non-uniformly to different lepton flavours or uniformly is examined. Interestingly, if
+leptoquark couplings to electrons and muons are indeed universal, then the U1 model
+parameter space that corresponds to the anomalous contribution should already
+accessible with Run-2 data in the non-resonant eµ channel. In the non-universal
+case, there is a significant exclusion in couplings, but not enough to reach regions
+that explain observed anomalies. We, therefore, examine the prospective sensitivity
+at the HL-LHC as well as of a 3 TeV future muon collider. For the vector leptoquark
+model, we find that a muon collider can probe all of the relevant parameter space at
+95% confidence with just 1 fb−1 data whereas R2 and S3 models can be excluded at
+95% with 5 fb−1 and 6.5 fb−1 luminosity respectively.
+∗ nishita.desai@tifr.res.in
+† amartya.sengupta@studenti.unipd.it
+arXiv:2301.01754v1  [hep-ph]  4 Jan 2023
+
+2
+I.
+INTRODUCTION
+An exciting development in recent years has been the measurement of ratios of de-
+cay widths in the semileptonic rare decays of B-mesons [1–4], hinting at lepton flavour-
+universality violation (LFV). The latest of these [1, 2] showed a measurement consistent
+with the SM for certain lepton universality, however, there remains a mismatch with the
+measured branching fraction of Bs → µ+µ− [3, 4] and in the angular distribution in the
+decay B → K∗µ+µ− [5, 6]. Unsurprisingly, this has led to a spirited effort to understand
+the source of the mismatch with the predictions of the Standard Model (SM) and to provide
+new physics explanations for it. In particular, there have been several dedicated studies that
+determine global fits to data in terms of effective field theoretic operators (see e.g. [7–11]).
+There has also been some effort to explain the anomalies in terms of new particles, notably
+with new vector bosons or leptoquarks [12–15]. The effects of the presence of such new
+particles can generally be seen in other observables besides the LFV ratios, and in partic-
+ular, in the high energy tails of certain distributions observable at the LHC. In this paper,
+we examine the expected effects of leptoquarks with minimally required properties to cause
+observed anomalies in the B-sector and report on current constraints and future prospects
+of their detection.
+We start by providing a bare-bones introduction to how the Effective Field Theoretic
+(EFT) framework is used and translated to the measurement of the high-energy observables
+that we examine in this paper. EFT provides a useful method to describe the low-energy
+physics processes in which the short-distance (i.e. high-energy or UV) physics is encapsulated
+in the Wilson coefficients whilst the rest of the long-distance physics is expressed in terms of
+effective operators with those having dimensions higher than four being suppressed by powers
+of an energy scale to maintain the mass dimension of each term in the Lagrangian. The
+analytic form of the Wilson coefficient can then be calculated by “matching” the expressions
+calculated from the EFT with the expressions from the full UV theory. We can use the
+published value by one of the multiple groups to translate the B-meson observations into
+best-fit values of the appropriate Wilson coefficients [7–11, 16]. We then match these values
+to the expressions derived from the leptoquark model under study and study the consequence
+of what that means on other production mechanisms at the LHC.
+The anomalies seen in the data fall into two categories — (1) in the neutral current sector
+
+3
+with b → s transitions, and (2) in the charged current sector with b → c transitions. In
+this work, we concentrate mainly on models that explain the first of these [17], however, it
+is known that one of the models we study viz. the U1 vector leptoquark can explain both
+simultaneously(see e.g. table 2. of [12])
+The relevant observations that motivate this work based on the full Run 1 and 2 dataset
+are shown in table II in the appendix. For completeness, we show both the pre-December
+2022 LHCb announcement [1, 2] numbers, as well as the latest measurements.
+The low-energy effective theory for the b → s flavour changing neutral current sector is
+described in terms of an effective Hamiltonian which can be written as
+Heff = −4GF
+√
+2 VtbV ∗
+ts
+� �
+Ci(µ)Oi(µ)
+�
+where Ci(µ) are the Wilson coefficients. The effective operators relevant to our study are
+Ol1l2
+9
+=
+e2
+(4π)2(¯sγµPLb)(¯l1γµl2),
+Ol1l2
+10 =
+e2
+(4π)2(¯sγµPLb)(¯l1γµγ5l2)
+(I.1)
+Multiple fitting studies have found that the operator whose Wilson coefficient shows sig-
+nificant deviation from the predicted SM value is the C9 and that the most likely discrepancy
+seems to be in the Cµ+µ−
+9
+coefficient. To stay consistent with the latest data, we use the
+Author (Year)
+Model Dependent Data Driven
+Ciuchini et al (2022) [7]
+[−1.25, −0.72]
+[−1.10, 1.05]
+Ciuchini et al (2019) [8]
+[−1.37, −1.05]
+[−1.47, −0.93]
+Alguer´o et al (2019) [9]
+[−1.15, −0.81]
+Alok et al (2019) [10]
+[−1.27, −0.91]
+Mahmoudi et al (2021) [11]
+[−1.07, −0.83]
+TABLE I. Best Fit values for the new physics contribution to the operator C9. The first of these
+contains the updated 2022 results. The fits taking into account angular distributions still favour
+a similar range as before the 2022 LHCb data release even though the overall best fit 1σ range is
+now consistent with the SM value of zero.
+
+4
+most recent best-fit results as reported in [7]. We shall use the best fit values that correctly
+give the angular correlations as well (the so called “model-dependent” fit). However, later
+in the paper when we examine future prospects, we also show the overlap with the fully
+agnostic data-driven fits. For an overview of the best-fit C9 values see table I. Currently, we
+proceed by using the value
+Cµ+µ−
+9
+= −0.98 ± 0.27,
+Multiple studies have also examined the leptoquark UV completion and calculated explicit
+expressions for Cµ+µ−
+9
+from each model. In this work, we use these expressions to investigate
+the LHC constraints on the couplings and mass of the leptoquarks.
+We make only the
+minimal assumptions, i.e. only the couplings that are necessary to give a contribution to
+the b → s anomalies is assumed to be non-zero. As we shall see, in each leptoquark model,
+the Wilson coefficients C9(10) depend on three parameters roughly as
+C9 ∼
+�y22 y32
+M
+�2
+where y22 is the sµ coupling, y32 is the bµ coupling and M is the mass of the leptoquark. We
+start by constraining (y22, y32, M) in other production modes without any further assump-
+tions on other leptoquark couplings. This results in the most conservative limits. In the case
+where there is no LFV, one would expect identical couplings of the leptoquark to electrons,
+i.e. y22 = y21 and y32 = y31. This would also lead to signatures with different flavored
+dileptons which often have much stronger constraints. These constraints are examined in
+section III. In the flavour universal case, the strongest limits on leptoquark masses will come
+from µ → e processes including µ → eγ [18] and µ → 3e[19] measurements. However, it
+might be possible that the effects of leptoquarks could be cancelled in loop-induced processes
+by the presence of other new particles. Studying direct leptoquark production at the LHC
+allows us to directly probe the lepton-universal case because the observed number of events
+in µµ, ee and µe channels will be correlated.
+Our paper is structured as follows: we start by listing out the model Lagrangian and
+the resulting Wilson coefficients for C9 in section II. We then examine the current LHC
+constraints in various search channels in section III and expected detection prospects of
+future colliders are calculated in section IV.
+
+5
+II.
+LEPTOQUARK MODELS
+Leptoquarks are bosons which carry both SU(2)L and colour SU(3) charges and therefore
+couple to both leptons and quarks. Given that we need to get the right contribution to
+Cµ+µ−
+9
+, this corresponds to a leptoquark that at a minimum couples to muons and to b and
+s quarks. There are three known leptoquark models that give the right kind of contribu-
+tion [12–14, 20], which we describe below. We use the standard names for the fields, viz. S3,
+R2 and U1 and the numbers in brackets that follow correspond to (n-plet of SU(3), n-plet of
+SU(2), U(1)Y hypercharge). Of these, S3 and R2 are scalar fields and U1 is a vector field.
+A.
+Scalar Leptoquark S3 (¯3, 3, 1/3)
+The first leptoquark model we consider is S3(¯3, 3, 1/3) which is a SU(2)L triplet of scalar
+leptoquark states with hypercharge 1/3. S3 is the only scalar leptoquark model that can
+simultaneously predict Rexp
+K∗ < RSM
+K∗ and Rexp
+K∗ < RSM
+K∗ at tree level [21–24]. The Lagrangian
+for the S3 model is
+LS3 = yij
+L ¯QC
+i iτ2(τkSk
+3)Lj + h.c.,
+(II.1)
+where Qi and Lj are SU(2)L doublet fermion fields corresponding to quarks and leptons of
+the ith( jth) generation respectively, τk are the generators of SU(2)L, and yij
+L stands for a
+Yukawa matrix for the left-handed fermions. The three triplet component states of S3 carry
+charges Q = −2/3, 1/3 and 4/3 respectively. Expanding out the SU(2)L components and
+referring to the leptoquarks as SQ
+3 , we get
+LS3 = −yij
+L ¯dC
+LiνLjS1/3
+3
+−
+√
+2yij
+L ¯dC
+LiℓLjS4/3
+3
++
+√
+2(V ∗yL)ij¯uC
+LiνLjS−2/3
+3
+− (V ∗yL)ij¯uC
+LiℓLjS1/3
+3
++ h.c.,
+(II.2)
+of which only the ¯dC
+LiℓLjS4/3
+3
+term contributes to O9. One can extract the Wilson coefficients
+for the b → sl−l+ decay [12–14, 20],
+Cℓ1ℓ2
+9
+= −Cℓ1ℓ2
+10
+=
+πv2
+VtbV ∗
+tsαem
+ybℓ1
+L (ysℓ2
+L )∗
+m2
+S3
+,
+(II.3)
+
+6
+B.
+Scalar Leptoquark R2 (3, 2, 7/6)
+The second case we consider is a weak doublet of scalar leptoquarks with hypercharge Y =
+7/6, i.e. R2 (3, 2, 7/6).[25] The most general Lagrangian describing the Yukawa interactions
+with R2 can be written as,
+LR2 = yij
+R ¯QilRjR2 − yij
+L ¯uRiR2iτ2Lj + h.c.,
+(II.4)
+where yL and yR are the Yukawa matrices corresponding to left- and right-handed lepton
+fields respectively. In terms of the components with RQ
+2 denoting each leptoquark state with
+charge Q, the Lagrangian can be written as
+LR2 = (V yR)ij¯uLiℓRjR5/3
+2
++ (yR)ij ¯dLiℓRjR2/3
+2
++ (yL)ij¯uRiνLjR2/3
+2
+− (yL)ij¯uRiℓLjR5/3
+2
++ h.c.
+(II.5)
+The tree-level contribution to the Wilson coefficients C9 through the term (yR)ij ¯dLiℓRjR2/3
+2
+amounts to
+Cℓ1ℓ2
+9
+= Cℓ1ℓ2
+10
+= −
+πv2
+2VtbV ∗
+tsαem
+ysℓ1
+R (ybℓ2
+R )∗
+m2
+R2
+,
+(II.6)
+C.
+Vector Leptoquark U1 (3, 1, 2/3)
+Finally, we describe the only vector leptoquark model considered in this paper, mainly
+because it has been the only model that could simultaneously explain both charged current
+and neutral current anomalies [12]. We consider the U1 (3, 1, 2/3) model which gives a single
+leptoquark state with charge 2/3. The most general Lagrangian consistent with the SM
+gauge symmetry allows couplings to both left-handed and right-handed fermions, namely
+LU1 = βij
+L ¯QiγµLjU µ
+1 + βij
+R ¯dRiγµℓRjU µ
+1 + h.c.,
+(II.7)
+with couplings βij
+L and βij
+R. The contributions to the left-handed couplings to the effective
+Lagrangian amount to
+Cℓ1ℓ2
+9
+= −Cℓ1ℓ2
+10
+= −
+πv2
+VtbV ∗
+tsαem
+βsℓ1
+L (βbℓ2
+L )∗
+m2
+U1
+,
+(II.8)
+
+7
+III.
+LHC LIMITS
+Our goal is to use published LHC data to simultaneously constrain the mass and Yukawa
+couplings of the leptoquarks. The Wilson coefficient C9 depends on three parameters roughly
+as
+Cℓ,ℓ
+9
+∼
+�y2ℓ y3ℓ
+M
+�2
+where yij refers to the leptoquark coupling between the ith generation of quark and jth
+generation lepton. This corresponds to Yukawa couplings for S3 and R2 models and the gauge
+coupling for the U1 model. Therefore, its possible to find a surface in the 3D parameter space
+that gives the required value of C9. However, most LHC search constraints are in principle
+only 2D — one coupling that determines the cross section of the final state and one mass.
+We, therefore, have several options in which to view the full constraints.
+Let us start with ℓ = 2 (i.e. µ) which contributes to Cµµ
+9 . To be able to independently
+constrain the two Yukawa couplings y22 and y32, we study three different cases — first
+setting only y22 non-zero (see figure 1, second setting only y32 non-zero (see figure 4) and
+third, setting both equal (see figure 5). Using the upper limits from the non-resonant dimuon
+search gives us an upper limit on y22 at each mass value. It is possible to also determine the
+minimal allowed value of y22 that is consistent with C9 by requiring y32 ≤ 1.
+Since the latest LHCb data seem to indicate that electrons and muons have identical
+behaviour, we can indeed also do a similar exercise with y21 and y31 which would contribute
+to Cee
+9 . Besides these, non-zero values of all four couplings (or even a single electron and
+a single muon coupling) — y21, y31, y22 and y32 can give signatures that have differently
+flavoured leptons in the final state, but without missing energy and therefore with no SM
+background.
+It should be noted that in the case where a single leptoquark state can couple to both
+electrons and muons, the strongest constraints on couplings and mass of course come from
+low energy processes in the µ → e sector [18, 19, 26]. However, it can still be an interesting
+exercise to directly probe the case where both yk1 and yk2 are non-zero. As we see in figure 2,
+this case is strongly constrained by the LHC, with the U1 model likely to be ruled out already
+with full run-2 data of 139 fb−1.
+Since multiple leptoquark states come from the same multiplet, they have identical mass
+and switching on a single coupling allows the production of multiple states. For calculating
+
+8
+the LHC limits, we allow the production of all leptoquark states and select only that fraction
+that decays into the final state selected for by the analysis being reinterpreted. For example,
+in the S3 case, if we look for pair production of leptoquark followed by decay of each into
+a muon and a jet by turning on y22 ̸= 0 alone, we allow both the production of pairs of
+S4/3
+3
+→ ¯sµ+ as well as pairs of S1/3
+3
+→ ¯cµ+. Our limits, therefore, are not identical to the
+simplified model limits that the experimental analysis publishes by producing only one state
+at a time, with 100% branching fraction into a certain channel. Similarly, when looking at
+dilepton distributions, we take into account, with interference, all leptoquark states in the
+t-channel that are allowed by non-zero couplings.
+A.
+Computational setup
+Since we examine the limits from dilepton searches which have been presented in the form
+of upper limits on generator-level cross sections with fiducial cuts, our computational setup
+is much simplified. We generate events using
+Madgraph5 amc@NLO [27] with the required
+fiducial cuts and do not need to perform further detector simulation. This approach has
+been proven to work well [28] and reproduces expected limits. For the UV models, we use
+the scalar leptoquark models for S3 and R2 described in [29] and for the vector leptoquark
+model for the U1 case, we use the model described in [30–32]. When more complicated
+functionality is required, we use Pythia8 [33] to shower, hadronize and apply the required
+kinematic cuts on events.
+B.
+Limits from resonant and non-resonant dilepton searches
+We re-interpreted both the dilepton resonance search with 139 fb−1 [34] and the non-
+resonant dilepton search at 139 fb−1 [35] from ATLAS. We find that the non-resonant search
+results in much stronger limits and we continue with this search for the rest of our study.
+The exclusive dilepton state can only be seen with a t-channel leptoquark exchange. It is
+possible to have a dilepton plus two jets from strong production of leptoquarks, however,
+this process does not depend on the leptoquark-fermion couplings and results in only a mass
+limit which we deal with in the next subsection. With the interference of SM Drell-Yan
+production of leptons with the t-channel leptoquark mediated production, one expects to
+
+9
+1000
+2000
+3000
+4000
+5000
+6000
+7000
+8000
+0.001
+0.005
+0.010
+0.050
+0.100
+0.500
+1
+MLQ [GeV]
+y2k
+Allowed
+S3
+C9 ⇒ y3 k > 1
+1000
+2000
+3000
+4000
+5000
+6000
+7000
+8000
+0.001
+0.005
+0.010
+0.050
+0.100
+0.500
+1
+MLQ [GeV]
+y2k
+Allowed
+R2
+C9 ⇒ y3 k > 1
+1000
+2000
+3000
+4000
+5000
+6000
+7000
+8000
+0.001
+0.005
+0.010
+0.050
+0.100
+0.500
+1
+MLQ [GeV]
+β2k
+Allowed
+U1
+C9 ⇒ β3 k > 1
+FIG. 1.
+Exclusion plots y2ℓ versus Mass of leptoquark for the S3 (top-left), R2 (top-right) and
+U1 models (bottom). The bright red regions at the top are disallowed from dimuon searches. The
+corresponding di-electron limit is the lighter line inside the red region. The solid regions at the
+bottom are from requiring perturbative couplings consistent with allowed C9. The vertical lines
+are mass limits from direct leptoquark pair production with the solid line corresponding to second
+generation leptons and the dotted corresponding to first generation. The limits correspond to 139
+fb−1 data.
+see a change in the shape of the dilepton invariant mass distribution mℓℓ where ℓ = µ or e.
+We apply the limits from the ATLAS non-resonant dilepton search by generating events
+using
+Madgraph5 amc@NLO according to fiducial cuts listed in [35] and using the 95% upper
+limits for the most conservative signal region called the “µ+µ− constructive signal region” (or
+
+10
+analogously the e+e− constructive signal region). The constructive signal region corresponds
+to the case where you expect signal events above the EW expectation, which is similar to
+our case. The experimental analysis uses LO signal shape to model the expected number of
+events and we therefore also do not use any NLO corrections. The upper limits are provided
+on the additional cross section above the expected SM Electro-Weak (EW) prediction in the
+cumulative signal region where mµ+µ− ≥ 2070 GeV (or me+e− ≥ 2200 GeV).
+As expected, the effect of having heavy new leptoquarks in t-channel dies down when
+either the leptoquark mass is too high or the Yukawa coupling is too small. To account for
+the interference correctly, we use the difference of the cross-section pp → ℓ+ℓ− with both
+leptoquark and EW bosons, and with only EW gauge bosons as our new physics contribution.
+The result is an excluded region near high Yukawa coupling values, with a larger range ruled
+out for smaller leptoquark masses. This is shown as a bright red region in figure 1. The
+highest allowed value of y2k is referred to as y2k max and can be used to further restrict what
+values of y3k are consistent with C9.
+Currently, there is one different flavour dilepton search [36] performed at 13 TeV, but
+with only 3.2 fb data analysed. Aside from cuts on pT of 65 and 50 GeV on electrons and
+muons respectively, there are requirements that missing energy be less than 25 GeV and
+mT < 50 GeV to remove contamination from W-boson production which we apply using
+Pythia 8.3 [33]. The expected background for meµ > 2 TeV is 0.02 ± 0.02. They see one
+event and interpreting it as a statistical fluctuation, set a limit on new physics cross section.
+We extrapolate the expected limits from this search at 139 fb−1. The limits on the eµ case
+for the U1 model can be seen in figure 2. The expected background at 139 fb−1 is 2.78
+events, resulting in an expected 95% upper limit of 0.0185 fb on production cross section
+times branching. As can be seen, the U1 model should be completely ruled out with 139
+fb−1 data. For results in the eµ channel for S3 and R2 models, refer to appendix C.
+C.
+Limits from leptoquark-pair production
+Direct limits on the mass of the leptoquark based on strong pair-production mode followed
+by the decay of each leptoquark into a lepton and a jet are presented in [37]. The limits are
+also presented on generator-level cross-section times branching fraction and can be applied
+directly to our model. The resulting limit is shown as a solid black vertical line. Since there
+
+11
+is no significant improvement in the limit from b-tagging, we use the general lepton+jet
+limits in all cases. When only yk2 is non-zero, i.e. the leptoquark decays to a muon and a
+jet, we obtain a mass limit for S3 leptoquark at 1774 GeV, for the R2 leptoquark at 1720
+GeV and the U1 leptoquark at 2309 GeV. For the case where the leptoquark decays into
+electron alone, we get a mass limit for S3 leptoquark at 1828 GeV, for the R2 leptoquark at
+1773 GeV and the U1 leptoquark at 2419 GeV.
+There is no direct limit on the case with an eµ final state in the published search, which
+if it existed, would give a far better exclusion simply because there is no irreducible SM
+background and the dominant background would be from mis-identification of leptons.
+D.
+Missing search: top FCNC decay
+Given the need for non-zero leptoquark coupling to the third generation of quarks, this
+also implies a coupling between the top quark and second generation leptons for both the
+S3 and U1 models. In the R2 case, the coupling is either CKM suppressed (in the case of
+left-handed) or entirely independent and therefore set to zero (in the right-handed case). It
+would therefore be possible to search directly for FCNC top decay via t → cµµ.
+Currently, there are no searches for t → cµ+µ− except for a t → cZ search which requires
+the dimuon mass to be within 15 GeV of the Z mass [38] and therefore is not directly
+applicable to our model. A similar measurement from CMS [39] is available from the 8 TeV
+run.
+The main background for a t → cµ+µ− search is from the SM production of t¯tµ+µ−
+via an off-shell Z or γ produced in association with t¯t. To remove contamination from on-
+shell Z, we apply a cut instead Mℓℓ > 105 which is outside the Z-mass window selected for
+by the t → cZ searches. Assuming the identification acceptances do not change, we can
+estimate the background for our proposed search using the data driven estimate presented
+in [38] (denoted by σBG,ATLAS).
+Since we have identical SM production modes for t¯tZ
+and t¯tµ+µ−, we assume that the generator level transfer factor between these processes is
+transmitted all the way to the final selection. The kinematic effect of changing the mℓℓ cut
+from |Mℓℓ − MZ| < 15 to Mℓℓ > 105 can be estimated at generator level and is encapsulated
+in a single number fℓℓ Also, we assume that the enhancement in production of t¯tZ in going
+
+12
+1000
+2000
+3000
+4000
+5000
+6000
+7000
+8000
+0.001
+0.005
+0.010
+0.050
+0.100
+0.500
+MLQ [GeV]
+β2k
+Allowed
+U1
+C9 ⇒ β3 K > 1
+1000
+2000
+3000
+4000
+5000
+6000
+7000
+8000
+0.001
+0.005
+0.010
+0.050
+0.100
+0.500
+MLQ [GeV]
+β3k
+C9 ⇒ β2 k > 1
+U1
+C9 ⇒ β2 k > β2 k max
+1000
+2000
+3000
+4000
+5000
+6000
+7000
+8000
+0.001
+0.005
+0.010
+0.050
+0.100
+0.500
+MLQ [GeV]
+β2k,3k
+C9 best fit
+U1
+FIG. 2.
+Limits for the Leptoquark Couplings versus mass for the U1 Model. The dilepton process,
+in this case, is pp → µe which does not exist in the SM. We, therefore, have strong limits even
+with 3.2 fb−1 data as published in [36]. The top-left panel shows limits on the coupling to second
+generation quarks with y22 = y21, the top-right panel on the coupling to third generation quarks
+with y32 = y31 and the bottom panel shows the case where all four couplings are equal. The green
+band shows the values corresponding to the best fit values of C9 The dotted line in this figure
+shows the expected limit after analysing full 139 fb−1 of run-2 data by ATLAS (only partial result
+is published so far). We see clearly that the universal scenario is likely already ruled out by run-2
+data.
+
+13
+from 13 TeV to 13.6 TeV (fE = σ13.6
+σ13 ) remains the same also for t¯tµ+µ−. Thus we have
+σBG(√s = 13.6) =
+σBG,ATLAS
+× fE × fℓℓ
+× σ(pp → t¯tµ+µ−; √s = 13)
+σ(pp → t¯tZ; √s = 13)
+(III.1)
+Using this, and the expected background cross section from ATLAS, we calculate an
+expected background of 7±2 events. Given that with the Z-window, the background is esti-
+mated at 119±10 events, this would correspond to over an order of magnitude improvement
+in the sensitivity to FCNC branching fraction of the top quark.
+IV.
+FUTURE PROSPECTS
+The best-fit value of the Wilson coefficients for operators that explain the b → s anomalies
+suggests a high suppression scale. Using equations (II.6), (II.3) and (II.8), we find that the
+required scale for both couplings set to one is 16183 GeV for the R2 case and 22887 GeV
+for the S3 and U1 cases. Naturally, resonantly producing a leptoquark of this mass scale is
+out of the question at the LHC. We, therefore, investigate both the expected reach of the
+LHC after the planned high-luminosity run and estimate a conservative reach for a muon
+collider with CM energy of 3 TeV [40–43]. To illustrate the highest sensitivity case, we
+choose y22 = y32 for this calculation. This also allows us to make a comment on the ability
+of the collider to explore the entire parameter space of interest. A summary of the expected
+reach of future colliders can be seen in figure 3
+A.
+LHC High-Lumi expected limits
+Projections for the HL-LHC are made with the luminosity of 3000 fb−1. From previous
+experience, we know that the improvements in limits scale with about the square root of
+luminosity. Using the expected number of signal and background events for the non-resonant
+dilepton search, we can probe effects of leptoquarks up to mass 5 TeV for the S3, 3 TeV for
+the R2 and 9.5 TeV for the U1 model. Conversely, we can probe coupling values as small as
+0.4 for S3, 0.55 for R2 and 0.15 for U1 models respectively at 1 TeV leptoquark mass. For
+
+14
+1000
+2000
+3000
+4000
+5000
+6000
+7000
+8000
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+MLQ [GeV]
+y22,32
+LHC13
+MuonC, 1/fb
+5σ
+2σ
+C9
+fit
+S3
+LHC13-Mass
+1000
+2000
+3000
+4000
+5000
+6000
+7000
+8000
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+MLQ [GeV]
+y22,32
+LHC13
+MuonC, 1/fb
+5σ
+2σ
+C9
+fit
+R2
+LHC13-Mass
+1000
+2000
+3000
+4000
+5000
+6000
+7000
+8000
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+MLQ [GeV]
+β22,32
+LHC13
+MuonC, 1/fb
+5σ
+2σ
+C9
+fit
+U1
+LHC13-Mass
+FIG. 3. Current and future reach in leptoquark coupling to muons with leptoquark mass for the S3
+Model (top-left), R2 Model (top-right) and U1 Model (bottom). The green region corresponds to
+the 1σ region given by global fit C9 values in the model-dependent case whereas the yellow is the
+data-driven 1σ region ([7], also see table I). The solid red region is the current 139 fb−1 limits with
+the dotted red line the expected reach after 3 ab−1 at the HL-LHC. The solid and dotted vertical
+lines correspond to mass limits from pair production again corresponding to the 139 fb−1 and 3
+ab−1 luminosity respectively. The blue region corresponds to the parameter space that can be
+discovered with a 5σ significance at a 3 TeV muon collider with 1 fb−1 whereas the orange region
+corresponds to the further region that can be probed at 95% confidence at the same collider. The
+U1 model can be fully excluded with just 1 fb−1 data. The S3 and R2 models can also be fully
+probed with 6.5fb−1 and 5fb−1 respectively.
+comparison, C9 best fit predicts a minimum value of coupling at 0.04, 0.06 and 0.04 for the
+three models when we set both couplings equal.
+The direct search limits from strong production are calculated in a similar way using the
+
+15
+published upper limits at 139/fb. We find that the HL-LHC can exclude leptoquark masses
+of 2.2 TeV for both the S3 and R2 case and 2.8 TeV for the U1 case for the leptoquark
+decaying into a muon and a jet and 2.3 TeV for both the S3 and R2 case and 2.9 TeV for
+the U1 case for the leptoquark decaying into an electron and a jet.
+B.
+Reach of a Future Muon Collider
+Estimating the reach of a future muon collider is more difficult since we do not currently
+have a detector configuration to be able to simulate a realistic analysis. However, taking
+lessons from the dilepton and dijet searches at the LHC, we know that a single-bin analysis
+with a high enough cut on the invariant mass provides a very reliable estimate of reach. We
+look at µ+µ− → jj as our signal. Obviously using b-tagging will be a further improvement
+that can pinpoint the underlying scenario. However, for this estimate, we just use untagged
+jets. Given that acceptance efficiencies of jets are expected to be similar for both signal and
+background events for a simple dijet search, we proceed with using just generator-level cross
+sections. A further advantage is the much reduced probability of extra initial state radiation
+jets from initial state muons (in sharp contrast to a pp machine).
+The main background from the SM comes from s-channel photon or Z exchange. In the
+presence of the leptoquark, another Feynman diagram with a t-channel leptoquark exchange
+needs to be taken into account. We look only at events with Mjj > 500 GeV. The SM-only
+cross section at LO is 5.96×10−2 pb which corresponds to a statistical error of about 8 events
+at a luminosity of 1 fb−1. Using this, we can calculate the parameter space corresponding to
+a 5σ discovery as well as regions that can be excluded at 2σ. They are shown in figure 3 as
+blue and orange regions respectively. In the U1 case, we see that a muon collider is capable
+of excluding the entire viable parameter space with 1 fb−1. To exclude the R2 and S3 models
+would need a luminosity of 6.5 fb−1 for S3 and 5 fb−1 for R2.
+V.
+SUMMARY AND CONCLUSIONS
+We examine the limits from direct collider searches on leptoquark models that are capable
+of explaining the anomalous measurements in the decays of B-mesons. We focus on three
+specific models — two scalar leptoquark models S3 and R2 and one vector leptoquark model
+
+16
+U1. Aside from limits on the mass of the leptoquarks (which can be pair-produced by strong
+interactions), it is possible to also constrain the couplings to fermions by looking at changes
+to the shape of the dilepton mass spectrum. Reinterpreting full Run-2 limits from the pair
+production and non-resonant dilepton searches by ATLAS experiment, we find that current
+mass limits are 1.77 TeV, 1.72 TeV and 2.3 TeV respectively for the three models. We can
+expect to reach up to 2.2 TeV for S3 and R2 and 2.8 TeV for the U1 respectively with the
+High-Luminosity LHC run.
+Effects of leptoquarks with couplings to muons can potentially be probed in a muon
+collider. Since there has been considerable interest in a future muon collider recently, we
+also estimate what the reach of the proposed 3 TeV muon collider would be for the three
+models in question. We find that with very minimal assumptions, S3, R2 and U1 models
+show significant deviation in dijet distributions that can be observable for the entire range
+of interest with less than 6 fb−1 data for all three models.
+ACKNOWLEDGEMENTS
+ND is supported by the Ramanujan Fellowship grant SB/S2/RJN-070 from the Department
+of Science and Technology of the Government of India.
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+21
+Appendix A: Relevant observables in the b → s sector
+Observable
+Experiment
+Theory (SM)
+RK[0.1,1.1]
+0.994 +0.090
+−0.082 (stat) +0.029
+−0.027 (syst) [2022] [1, 2]
+1.00 ± 0.01 [44])
+RK∗[0.1,1.1]
+0.927 +0.093
+−0.087 (stat) +0.036
+−0.035 (syst) [2022] [1, 2]
+1.00 ± 0.01 [44])
+RK[1.1,6]
+0.949 +0.042
+−0.041 (stat) +0.022
+−0.022 (syst) [2022] [1, 2]
+1.00 ± 0.01 [44])
+RK∗[1.1,6]
+1.027 +0.072
+−0.068 (stat) +0.027
+−0.026 (syst) [2022] [1, 2]
+1.00 ± 0.01 [44])
+R[0.045,1.1]
+K∗
+0.66+0.11
+−0.07 ± 0.03 [2021] [45]
+0.906 ± 0.028 [44]
+R[1.1,6.0]
+K∗
+0.69+0.11
+−0.07 ± 0.05 [2021] [45]
+1.00 ± 0.01 [44]
+R[1.1,6.0]
+K
+0.846+0.042+0.013
+−0.039−0.012 [2021] [46]
+1.00 ± 0.01 [44]
+B(Bs → µ+µ−)
+(2.85+0.32
+−0.31) × 10−9 [3, 4]
+(3.66 ± 0.14) × 10−9[47])
+P ′
+5 in B → K(∗) l+ l−
+[5, 48, 49]
+[6, 50]
+TABLE II. A summary of the most relevant experimental results and SM predictions for the
+observables in b → s sector.
+
+22
+Appendix B: Limits on leptoquark couplings to third generation quarks y3k.
+1000
+2000
+3000
+4000
+5000
+6000
+7000
+8000
+0.005
+0.010
+0.050
+0.100
+0.500
+1
+MLQ [GeV]
+y3k
+C9 ⇒ y2 k > 1
+S3
+C9 ⇒ y2 k > y2 k max
+1000
+2000
+3000
+4000
+5000
+6000
+7000
+8000
+0.001
+0.005
+0.010
+0.050
+0.100
+0.500
+1
+MLQ [GeV]
+y3k
+C9 ⇒ y2 K > 1
+R2
+C9 ⇒ y2 K > y2 K max
+1000
+2000
+3000
+4000
+5000
+6000
+7000
+8000
+0.001
+0.005
+0.010
+0.050
+0.100
+0.500
+1
+MLQ [GeV]
+β3k
+C9 ⇒ β2 k > β2 k max
+U1
+FIG. 4.
+Exclusion plots y3ℓ versus Mass of leptoquark for the S3 (top-left), R2 (top-right) and
+U1 models (bottom). The solid regions at the bottom are from requiring perturbative couplings
+consistent with allowed C9. The darker region is inconsistent with the observed upper limits on y2k
+in figure 1. The vertical lines are mass limits from direct leptoquark pair production with the solid
+line corresponding to second generation leptons and the dotted corresponding to first generation.
+The limits correspond to 139 fb−1 data.
+
+23
+1000
+2000
+3000
+4000
+5000
+6000
+7000
+8000
+0.001
+0.005
+0.010
+0.050
+0.100
+0.500
+1
+MLQ [GeV]
+y2k,3k
+C9 best fit
+S3
+1000
+2000
+3000
+4000
+5000
+6000
+7000
+8000
+0.001
+0.005
+0.010
+0.050
+0.100
+0.500
+1
+MLQ [GeV]
+y2k,3k
+C9 best fit
+R2
+1000
+2000
+3000
+4000
+5000
+6000
+7000
+8000
+0.001
+0.005
+0.010
+0.050
+0.100
+0.500
+1
+MLQ [GeV]
+β2k,3k
+C9 best fit
+U1
+FIG. 5.
+Exclusion plots in the limited case of y2ℓ = y3ℓ versus Mass of leptoquark for the S3
+(top-left), R2 (top-right) and U1 models (bottom).
+The solid red region at the top are limits
+from non-resonant dilepton searches in µ+µ−. The lighter lines inside this region correspond to
+subleading limits from the similar e+e− search.
+The vertical lines are mass limits from direct
+leptoquark pair production with the solid line corresponding to second generation leptons and the
+dotted corresponding to first generation. The limits correspond to 139 fb−1 data. The green band
+is the region that corresponds to the coefficient C9 within one sigma of best fit to data.
+–
+
+24
+Appendix C: Limits on S3 and R2 model parameters in the Lepton Flavour Universal
+case
+1000
+2000
+3000
+4000
+5000
+6000
+7000
+8000
+0.001
+0.005
+0.010
+0.050
+0.100
+0.500
+1
+MLQ [GeV]
+y2k
+Allowed
+S3
+C9 ⇒ y3 K > 1
+1000
+2000
+3000
+4000
+5000
+6000
+7000
+8000
+0.005
+0.010
+0.050
+0.100
+0.500
+1
+MLQ [GeV]
+y3k
+C9 ⇒ y2 k > 1
+S3
+C9 ⇒ y2 k > y2 k max
+1000
+2000
+3000
+4000
+5000
+6000
+7000
+8000
+0.001
+0.005
+0.010
+0.050
+0.100
+0.500
+1
+MLQ [GeV]
+y2k,3k
+C9 best fit
+S3
+FIG. 6. Limits on the leptoquark couplings via the process p p → µ e in the case of flavour universal
+couplings to electrons and muons for the S3 Model.
+
+25
+1000
+2000
+3000
+4000
+5000
+6000
+7000
+8000
+0.001
+0.005
+0.010
+0.050
+0.100
+0.500
+1
+MLQ [GeV]
+y2k
+Allowed
+R2
+C9 ⇒ y3 K > 1
+1000
+2000
+3000
+4000
+5000
+6000
+7000
+8000
+0.001
+0.005
+0.010
+0.050
+0.100
+0.500
+1
+MLQ [GeV]
+y3k
+C9 ⇒ y2 k > 1
+R2
+C9 ⇒ y2 k > y2 k max
+1000
+2000
+3000
+4000
+5000
+6000
+7000
+8000
+0.001
+0.005
+0.010
+0.050
+0.100
+0.500
+1
+MLQ [GeV]
+y2k,3k
+C9 best fit
+R2
+FIG. 7. Limits on the leptoquark couplings via the process p p → µ e in the case of flavour universal
+couplings to electrons and muons for the R2 Model.
+

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