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samples/texts/102047/page_11.md

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+[DM2] W. Dahmen and C. A. Micchelli, *On the local linear independence of translates of a box spline*, Studia Math., **82** (1985), 243–263.
+
+[DM3] —, *Multivariate E-splines*, Advances in Math., **76** (1989), 33–93.
+
+[DR1] N. Dyn and A. Ron, *Local approximation by certain spaces of multivariate exponential-polynomials, approximation order of exponential box splines and related interpolation problems*, Trans. Amer. Math. Soc., **319** (1990), 381–404.
+
+[DR2] —, *On multivariate polynomial interpolation*, in *Algorithms for Approximation II*, J. C. Mason, M. G. Cox (eds.), Chapman and Hall, London 1990, 177–184.
+
+[G] J. A. Gregory, *Interpolation to boundary data on the simplex*, CAGD, 2 (1985), 43–52.
+
+[J] R. Q. Jia, *A dual basis for the integer translates of an exponential box spline*, preprint 1988.
+
+[R] A. Ron, *Relations between the support of a compactly supported function and the exponential-polynomials spanned by its integer translates*, Constructive Approx., **5** (1989), 297–308.
+
+Received April 17, 1989. The first author was supported by the National Science Foundation under Grant No. DMS-8701275 and by the United States Army under Contract No. DAAL03-87-K-0030.
+
+UNIVERSITY OF WISCONSIN
+MADISON, WI 53706
+
+TEL-AVIV UNIVERSITY
+TEL-AVIV, ISRAEL
+
+AND
+
+UNIVERSITY OF WISCONSIN
+MADISON, WI 53706

samples/texts/102047/page_14.md

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+We assume that $X$ spans all of $\mathbb{R}^s$. Then the only point common to all $h \in \mathbb{H}(X)$ is 0, and consequently, the variety of $I^X$ (i.e., the set of common zeros of all the polynomials in $I^X$) consists of 0 alone:
+
+$$ \mathcal{V}_{I^X} = \{\emptyset\}. $$
+
+This implies that the codimension of $I^X$ in the space $\Pi$ of all polynomials in $s$ variables (i.e., the dimension of the quotient space $\Pi/I^X$) is finite, and that its kernel $I^X \perp$ is a finite-dimensional polynomial space, whose dimension equals the codimension of $I^X$. Moreover, $I^X \perp$ is *stratified*, i.e., spanned by its homogeneous elements, since $I^X$ is generated by homogeneous polynomials. [BR2] is a ready reference for these known facts.
+
+Here, to recall the definition, the kernel $I \perp$ of an ideal $I$ is the set
+
+$$ (2.2) \qquad \{f \in \mathcal{P}'(\mathbb{R}^s) : p(D)f = 0, \forall p \in I\} $$
+
+of all distributions annihilated by the set of differential operators induced by $I$. In particular, since $I^X$ is generated by
+
+$$ p_h = \langle h^\perp, \cdot \rangle^{(X\setminus h)}, \quad h \in \mathbb{H}(X), $$
+
+$I^X \perp$ consists of the solutions $f$ of the system of linear differential equations
+
+$$ (2.3) \qquad (D_{h^\perp})^{*(X\setminus h)} f = 0, \quad \forall h \in \mathbb{H}(X). $$
+
+This section is devoted to a proof of the fact that $I^X \perp$ equals the polynomial space
+
+$$ (2.4) \qquad \mathcal{P}(X) := \operatorname{span}\{p_V : V \subset X, \operatorname{span}(X\setminus V) = \mathbb{R}^s\}, $$
+
+with
+
+$$ p_V := \prod_{v \in V} \langle v, \cdot \rangle. $$
+
+In the proof, the multiset
+
+$$ (2.5) \qquad \mathbb{B}(X) := \{B \subset X : B \text{ invertible}\} $$
+
+of all bases contained in $X$ plays an important role. We use the abbreviation
+
+$$ (2.6) \qquad b(X) := \# \mathbb{B}(X). $$
+
+(2.7) **THEOREM.** The kernel $I^X \perp$ of $I^X$ coincides with $\mathcal{P}(X)$, and
+
+$$ (2.8) \qquad \dim I^X \perp = b(X). $$
+
+The theorem follows from the following three lemmata.

samples/texts/102047/page_16.md

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+with $\operatorname{span} X' = \mathbb{R}^s$, and consider the map
+
+$$T: I^{X'} \to \prod_{h \in \mathbb{H}(X')} P_h : q \mapsto (p_h(D)q)_{h \in \mathbb{H}(X')},$$
+
+where $p_h := p_{h, X'} = \langle h^\perp, \cdot \rangle^{(X'\setminus h)}$ are the generators of $I^{X'}$ and
+$P_h := p_h(D)(I^{X'}\perp)$. Then
+
+$$\dim I^{X'} \leq \dim I^{X'} \perp + \sum_{h \in \mathbb{H}(X')} \dim P_h,$$
+
+since
+
+$$\ker T = (I^{X'} \perp) \cap (I^{X} \perp) \subseteq I^{X'} \perp.$$
+
+Consequently, by induction,
+
+$$\dim(I^{X'} \perp) \leq b(X') + \sum_{h \in \mathbb{H}(X')} \dim P_h.$$
+
+This proves that
+
+$$\dim I^{X'} \perp \leq b(X),$$
+
+provided we can prove that
+
+$$\sum_{h \in \mathbb{H}(X')} \dim P_h \leq \#\{B \in \mathcal{B}(X) : x \in B\}.$$
+
+In particular, it is sufficient to prove that, for all $h \in \mathbb{H}(X')$,
+
+$$ (2.12) \qquad P_h \subset I^{X_h} \perp $$
+
+with
+
+$$X_h := (X \cap h) \cup x.$$
+
+For, (2.12) implies that
+
+$$\dim P_h \leq \dim I^{X_h} \perp \leq b(X_h),$$
+
+(the last inequality by induction), while
+
+$$\sum_{h \in \mathbb{H}(X')} b(X_h) = \#\{B \in \mathcal{B}(X) : x \in B\}.$$
+
+The claim (2.12) is trivial in case $x \in h$, since then $X'\setminus h = X\setminus h$,
+and therefore $p_{h,X'} = p_{h,X}$ and so $P_h = \{0\}$ in that case.
+
+We now prove (2.12) for the contrary case, i.e., the case when $x \notin h$. We have to show that for every $k \in \mathbb{H}(X_h)$
+
+$$ (2.13) \qquad p_{k, X_h}(D)P_h = \{0\}, $$

samples/texts/102047/page_17.md

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+with
+
+$$p_{k, X_h} = (k^{\perp}, \cdot)'^{(X_h \setminus k)}.$$
+
+If $x \notin k$, then $k = h$ and (since $X_h \setminus h = \{x\})$ $p_{k, X_h}$ is a linear polynomial, and there is nothing to prove since then $p_{k, X_h} p_{h, X'} = p_{h, X'}$, while $p_{h, X'}$ annihilates $I^X \perp$ by definition of $I^X \perp$. For the contrary case that $x \in k$, we need to prove that
+
+$$ (2.14) \qquad D_{k^\perp}^m D_{h^\perp}^{A-m} (I^X \perp) = 0, $$
+
+with
+
+$$ m := \#(X_h \setminus k), \quad A - m := \#(X' \setminus h). $$
+
+For this, it is sufficient to show that
+
+$$ (2.15) \qquad K^m H^{A-m} \in \text{ideal}\{L^{a(l)}\}_l, $$
+
+with $l$ running over all hyperplanes spanned by elements of $X$ and containing $k \cap h$, and with
+
+$$ K := \langle k^{\perp}, \cdot \rangle, \quad H := \langle h^{\perp}, \cdot \rangle, \quad L := \langle l^{\perp}, \cdot \rangle, \\ a(l) := \#(X \setminus l). $$
+
+For, (2.14) follows from (2.15) since each generator $L^{a(l)}$ of the ideal in (2.15) appears also in the defining set (2.1) of generators of $I^X$, hence annihilates $I^X \perp$.
+
+We prove (2.15) by writing each polynomial $L$ as a linear combination of the linear polynomials $K$ and $H$, thereby obtaining the general homogeneous element of degree $A$ of the ideal in (2.15) in the form
+
+$$ \sum_l (\alpha K + \beta H)^{a(l)} r_l, $$
+
+with $r_l$ a homogeneous polynomial of degree
+
+$$ \mu(l) := A - a(l), $$
+
+all $l$. We then show that the resulting linear system (for the coefficients of the various $r_l$) has a solution by showing that its coefficient matrix is the transpose of the matrix which occurs in (univariate polynomial) Hermite interpolation.
+
+Here are the details.
+
+Since each $l$ contains $k \cap h$, $l^\perp$ can be written uniquely as a linear combination of $k^\perp$ and $h^\perp$. We find it convenient in the sequel to have the weights in this linear combination sum to 1, i.e., to have $l^\perp$ in the affine hull of $k^\perp$ and $h^\perp$. This we can achieve by first

samples/texts/102047/page_18.md

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+choosing the (signed) magnitudes of the nonzero vectors $k^{\perp}$ and $h^{\perp}$ so that their difference is not perpendicular to any of the finitely many $l$. Then, for each $l$, we choose the nonzero vector $l^{\perp}$ to lie on the line through $k^{\perp}$ and $h^{\perp}$, i.e., so that
+
+$$l^{\perp} := (1 - \beta)k^{\perp} + \beta h^{\perp}$$
+
+for some $\beta = \beta_l$. Then $L = (1 - \beta)K + \beta H$; hence
+
+$$L^a = \sum_j K^j H^{a-j} B_j^a(\beta),$$
+
+with
+
+$$B_j^r(t) := \binom{r}{j} (1-t)^j t^{r-j}$$
+
+the polynomials appearing in the Bernstein form. Since $DB_j^r = r(B_{j-1}^{r-1} - B_{j-1}^{r-1})$, we have more generally
+
+$$((a+i)!/a!)(H-K)^i L^a = \sum_j K^j H^{a+i-j} D^i B_j^{a+i}(\beta).$$
+
+This means that we have available in our ideal an expression of the form
+
+$$\sum_l \sum_{i=0}^{\mu(l)} (H-K)^i L^{A-i} c(l, i) = \sum_j K^j H^{A-j} \sum_l \sum_{i=0}^{\mu(l)} D^l B_j^A(\beta) c(l, i)$$
+
+to match the monomial $K^m H^{A-m}$. Such a match is possible provided the linear system of conditions imposed thereby on the coefficients $c(l, i)$ is solvable.
+
+We begin the proof that this linear system is indeed solvable by showing that its coefficient matrix is square. With the abbreviation $k' := h \cap k$, we compute
+
+$$m = \#(X_h \setminus k) = \#((X \cap h) \setminus k') = \#((X' \cap h) \setminus k'),$$
+
+and therefore
+
+$$ (2.16) \qquad A = \#((X' \cap h) \setminus k') + \#(X' \setminus h) = \#(X' \setminus k'). $$
+
+Also,
+
+$$ 
+\begin{aligned}
+\mu(l) &= A - a(l) = \#(X' \setminus k') - \#(X \setminus l) = -1 + \#(X \setminus k') - \#(X \setminus l) \\
+&= \#((X \cap l) \setminus k') - 1.
+\end{aligned}
+$$
+
+Therefore, the number of unknowns is
+
+$$\sum_l (\mu(l) + 1) = \sum_l \#((X \cap l) \setminus k') = \#(X \setminus k') = A + 1,$$

samples/texts/102047/page_19.md

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+i.e., equal to the number of equations. Here, the sums are over all $l \in H(X)$ which contain $k'$, which implies that $X \setminus k'$ is the disjoint union of the sets $(X \cap l) \setminus k'$ and so justifies the second equality.
+
+Now organize the unknowns by $l$ and, within $l$, by $i=0, \dots, \mu(l)$, and order the equations by $j=0, \dots, A$. Then the matrix consists, more precisely, of one block of columns for each $l$, with the $i$th column (in the block for $l$) containing the value at $\beta = \beta_i$ of the $i$th derivative of all the polynomials $B_j^A$, $j=0, \dots, A$, $i=0, \dots, \mu(l)$. Hence our matrix is the transpose of the matrix which occurs in the linear system for the determination of the Bernstein form of the polynomial in $\Pi_A$ which agrees with some function $(\mu(l)+1)$-fold at $\beta = \beta_i$, all $l$. Since such univariate Hermite interpolation is correct (since $\beta_i \neq \beta_{i'}$ for $l \neq l'$), the invertibility of our matrix follows. $\square$
+
+We note that (2.7) Theorem now allows us to conclude that all inequalities appearing in its proof must be equalities. This implies, e.g., that
+
+$$I^{X'} \subset I^X \text{ whenever } X' \subset X,$$
+
+and that $p_{h,X'}(D)(I^X \perp) = I^{h \cup X} \perp$. Another immediate consequence is the following
+
+(2.17) COROLLARY [DM3], [DR1].
+
+$$\dim \mathcal{P}(X) = b(X).$$
+
+Furthermore,
+
+$$ (2.18) \qquad d(X) := \min\{\#(X\setminus h) : h \in \mathbb{H}(X)\} $$
+
+is the least degree of the generators of $I^X$; hence, since $I^X \perp = \mathcal{P}(X)$,
+we have the following.
+
+(2.19) COROLLARY [DR1]. With $d(X)$ as in (2.18),
+
+$$\Pi_{<d}(X) \subset \mathcal{P}(X),$$
+
+but
+
+$$\Pi_d(X) \not\subset \mathcal{P}(X).$$
+
+By its definition, $I^X \perp$ is the set of all polynomials $p \in \Pi$ that satisfy the following condition:
+
+(2.20) Condition. For every $h \in H(X)$ and $v \in \mathbb{R}^s$, $p|_{v+h\perp} \subset \Pi_{\#(X\setminus h)-1}$ (with $h\perp$ the subspace orthogonal to $h$).

samples/texts/102047/page_20.md

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+On the other hand, every $p \in \mathcal{P}(X)$ satisfies an apparently stronger condition (cf. [DR2; Prop. 1]):
+
+(2.21) Condition. For every subspace $M$ of $\mathbb{R}^s$ which is spanned by elements from $X$, and for every $v \in \mathbb{R}^s$,
+
+$$p|_{v+M^\perp} \in \Pi_{\#(X\setminus M)-\dim M^\perp}.$$
+
+Hence we conclude the following from (2.7) Theorem:
+
+(2.22) COROLLARY. The space $\mathcal{P}(X)$ is characterized either by (2.20) Condition or by (2.21) Condition. In particular, these two conditions are equivalent on $\Pi$.
+
+(2.22) Corollary verifies the claim made in [DR2; Remark 2] that (2.21) Condition characterizes $\mathcal{P}(X)$. In case $X$ consists of $N$ repetitions of $s+1$ vectors $Y \subset \mathbb{R}^s$ in general position, the characterization of $\mathcal{P}(X)$ by (2.21) Condition has been proved in [G] by other methods.
+
+**3. An associated polynomial interpolation problem.** Here we identify certain exponential spaces $H$ whose corresponding “limit at the origin” $H_\downarrow$ coincides with $\mathcal{P}(X)$ (for an appropriate choice of $X$), and use this identification in the solution of an associated interpolation problem. The map
+
+$$ (3.1) \qquad H \mapsto H_\downarrow, $$
+
+which associates with every finite-dimensional space of entire functions a homogeneous space of polynomials of the same dimension, has been introduced and studied in [BR1] in the context of a multivariate polynomial interpolation problem, and has been discussed as well in [BR2] in the context of kernels of polynomial ideals. To begin with, we recall the definition of $H_\downarrow$ and review some of the results of [BR1, 2] needed here. Then we discuss a certain interpolation problem and its relation to $\mathcal{P}(X)$.
+
+Given a function $f \neq 0$ analytic at the origin, we write its power series expansion at the origin in the form
+
+$$ f = f_0 + f_1 + f_2 + \dots, $$
+
+where, for each $j$, $f_j$ is a homogeneous polynomial of degree $j$, and define $f_\downarrow := f_k$ with $k = \min\{j: f_j \neq 0\}$; i.e., $f_\downarrow$ is the first non-trivial homogeneous polynomial in the power expansion of $f$. Using

samples/texts/102047/page_4.md

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+Assume now that $X$ is *unimodular*, i.e., the columns of $X$ are from $\mathbb{Z}^s\setminus0$ and every $B \in \mathbb{B}(X)$ has determinant $\pm1$. For such unimodular $X$, the observations made in [R; §4] (especially before the proof of Theorem 4.1 and in the proof of Corollary 4.2) confirm the existence, for each $h \in \mathbb{H}(X)$, of consecutive integers $c_{h,j}$ so that $\nu_X$ is contained in the union of the hyperplanes
+
+$$ \langle h^{\perp}, \cdot \rangle = c_{h,j}, \quad j = 1, \dots, \#X\backslash h. $$
+
+Indeed, fixing $h \in \mathbb{H}(X)$ and taking the normal $h^\perp$ to be a relatively prime integer vector implies that $\langle h^\perp, x \rangle = 1$ for every $x \in X\backslash h$ since $X$ is unimodular. By choosing the signs of the columns of $X$ appropriately (which amounts to a shift in $M_X$), we can achieve that $\langle h^\perp, x \rangle > 0$ for all $x \in X\backslash h$; hence
+
+$$ c_h := \sum_{x \in X \backslash h} \langle h^{\perp}, x \rangle = \#X \backslash h. $$
+
+Consequently, with $z$ chosen so as to satisfy $c_h - 1 < \langle h^\perp, z \rangle < c_h$, $\nu_X(z)$ must lie in the union of the hyperplanes
+
+$$ \langle h^{\perp}, \cdot \rangle = j, \quad j = 0, \dots, c_h - 1. $$
+
+Moreover, $\#\nu_X = \dim \mathcal{P}(X)$, because of (2.17) Corollary and the following.
+
+(3.11) **RESULT [DM2].** If $X$ is unimodular, then
+
+$$ \#\nu_X = b(X). $$
+
+This establishes the following theorem.
+
+(3.12) **THEOREM.** If $X$ is unimodular, then $(\exp_{\nu_X})_\downarrow = \mathcal{P}(X)$. In particular, $\nu_X$ is correct for $\mathcal{P}(X)$, and $\mathcal{P}(X)$ is of least degree among all polynomial spaces for which $\nu_X$ is correct.
+
+Note that only the inequality $\#\nu_X \ge b(X)$ was needed in the proof of (3.12) Theorem. As a matter of fact, the converse inequality is a consequence of the theorem.
+
+**4. The duality between $\mathcal{H}(X)$ and $\mathcal{P}(X)$.** The ideal $I_X$ and its kernel $\mathcal{P}(X)$ are intimately related to another ideal $I_X$ and its kernel $\mathcal{H}(X)$, which play a fundamental role in the theory of box splines. In the following, we review some of the basics about $I_X$ and $\mathcal{H}(X)$ and draw several connections between the two settings.

samples/texts/102047/page_6.md

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+polynomials, we may assume that this $p$ is homogeneous. But then
+$p^*(f) = p^*(f_\downarrow) \neq 0$, showing that the linear map $H \to P^*: f \mapsto (p \mapsto p^*(D)f(0))$ is 1-1; hence $P \to H^*: p \mapsto p^|_H$ is onto. Since $\dim H = \dim H_\downarrow$, and $\dim H_\downarrow = \dim P$ by assumption, the theorem follows. $\square$
+
+We note that the converse of (4.3) Theorem does not hold, in gen-
+eral. For, it is easy to make up a nonhomogeneous polynomial space
+$H$ together with a homogeneous $P$ dual to it, for which the condi-
+tions $\dim(\Pi_j \cap P) = \dim(\Pi_j \cap H_\downarrow)$, all $j$, fail to hold, while these
+conditions are necessary for $P$ and $H_\downarrow$ to be dual, according to the
+following proposition of use later.
+
+(4.4) PROPOSITION. If the homogeneous polynomial spaces Q and R are dual to each other, then
+
+$$
+(4.5) \quad \dim(\Pi_j \cap Q) = \dim(\Pi_j \cap R)
+$$
+
+for all $j$.
+
+*Proof*. Indeed, if (4.5) is violated for some (minimal) $j$ and, say, $\dim(\Pi_j \cap Q) > \dim(\Pi_j \cap R)$, then there exists a *homogeneous* polynomial $q \in Q$ of degree $j$ for which $q^*$ vanishes on all homogeneous polynomials in $R$ of degree $j$, and hence vanishes on all of $R$, in contradiction to the duality between $Q$ and $R$. $\square$
+
+With this, the meaning of the following result is clear.
+
+(4.6) **RESULT** [DM3]¹, [DR1]. *The polynomial spaces *$\mathcal{P}(X)$* and *$\mathcal{H}(X)$* are dual to each other.
+
+In (3.12) Theorem, the space $\mathcal{P}(X)$ has been identified as the least
+space for certain interpolation problems. In [BR2] the space $\mathcal{H}(X)$
+has been identified as the least space for other interpolation prob-
+lems. We now make use of the duality between $\mathcal{P}(X)$ and $\mathcal{H}(X)$
+to connect $\mathcal{H}(X)$ with the interpolation problems associated with
+$\mathcal{P}(X)$ and vice versa. As a preparation, we procure a class of spaces-
+whose corresponding least space is $\mathcal{H}(X)$ in much the same way in
+which we obtained suitable exponential spaces $\exp_{\nu_x}$ whose least is
+$\mathcal{P}(X)$: We perturb the linear factors of the set of generators for the
+
+¹The authors in [DM3] attribute the result to Hakopian.

samples/texts/102047/page_9.md

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+the following
+
+(5.2) COROLLARY. Let $X$ be a unimodular set of vectors. Then there exists a function (actually many) $\psi = \psi_X$ which is supported in $\Omega_X$ and satisfies
+
+$$\Pi(\psi) = \mathcal{P}(X).$$
+
+The above corollary provides no information about the smoothness
+of the compactly supported $\psi$. Yet, it is known [BH2] that, at least
+for the special case of the three-direction mesh with equal multiplic-
+ities, no piecewise-$\mathcal{P}(X)$ function supported on $\Omega_X$ can match the
+smoothness of the corresponding box spline $M_X$.
+
+The box spline $M_X$ is a smooth function supported on $\Omega_X$. Hence,
+one may hope that there exist functions $\phi$ supported on $\Omega_X$ which
+are less smooth than $M_X$, yet their corresponding $\Pi(\phi)$ is "better"
+in the sense that it contains some of the polynomials of lower degrees
+which were missing in $\Pi(M_X) = \mathcal{H}(X)$. [R; Cor. 4.2] gives a partial
+negative answer to that hope by showing that for a unimodular $X$ and
+a function $\phi$ supported in $\Omega_X$, if $\hat{\phi}(0) \neq 0$ and $\Pi_j \subset \Pi(\phi)$ for some
+$j$, then $\Pi_j \subset \mathcal{H}(X) = \Pi(M_X)$. The following result improves that
+corollary.
+
+(5.3) COROLLARY. Let $X$ be a unimodular set of directions, $M_X$
+the corresponding box spline. Let $\phi$ be a compactly supported function
+satisfying
+
+$$\operatorname{supp} \phi \subset \operatorname{supp} M_X,$$
+
+and
+
+$$\hat{\phi}(0) \neq 0.$$
+
+Then, for each $j$,
+
+$$\dim(\Pi_j \cap \Pi(\phi)) \leq \dim(\Pi_j \cap \Pi(M_X)).$$
+
+*Proof.* Let $\nu$ be one of the sets $\nu_X(z)$ associated with $X$. By (3.12) Theorem, $(\exp_\nu)_\downarrow = \mathcal{P}(X)$. Now we may apply (5.1) Result to conclude that $\nu$ is total for $\Pi(\phi)$, and hence $\Pi(\phi)$ can be extended to a space $Q$ for which $\nu$ is correct. Since $\mathcal{P}(X)$ is the least space of $\exp_\nu$, it satisfies the least degree property (3.4), thus we conclude that for every $j$
+
+$$ (5.4) \quad \dim(\Pi_j \cap \Pi(\phi)) \le \dim(\Pi_j \cap Q) \le \dim(\Pi_j \cap \mathcal{P}(X)). $$

samples/texts/1072043/page_17.md

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+### 6.2.3 Power factor, apparent power, active power and reactive power
+
+The power factor, the apparent power S (VA), the active power P (W), and the reactive power Q (Var) are related through the equations
+
+$$PF = \cos \phi = \frac{P(W)}{S(VA)} \qquad (21)$$
+
+$$S = VI' \qquad (22)$$
+
+$$P = S \cos \phi \qquad (23)$$
+
+$$Q = S \sin \phi \qquad (24)$$
+
+### 6.2.4 Harmonic calculation
+
+Total harmonic distortion of voltage ($THD_v$) and current ($THD_i$) can be calculated by the Equations 25 and 26, respectively.
+
+$$\%THD_{i} = \frac{\sqrt{\sum_{h=2}^{\infty} I_{h(rms)}^{2}}}{I_{1(rms)}} \times 100\% \qquad (25)$$
+
+$$\%THD_{v} = \frac{\sqrt{\sum_{h=2}^{\infty} V_{h(rms)}^{2}}}{V_{1(rms)}} \times 100\% \qquad (26)$$
+
+Where $V_h (\text{rms})$ is RMS value of h th voltage harmonic, $I_h (\text{rms})$ RMS value of h th current harmonic, $V_1 (\text{rms})$ RMS value of fundamental voltage and $I_1 (\text{rms})$ RMS value of fundamental current
+
+<table><thead><tr><th rowspan="2">Parameter</th><th colspan="3">Steady state FVHC condition</th><th colspan="3">Transient step down condition</th></tr><tr><th>Experimental</th><th>Modeling</th><th>% Error</th><th>Experimental</th><th>Modeling</th><th>% Error</th></tr></thead><tbody><tr><td>V<sub>rms</sub> (V)</td><td>218.31</td><td>218.04</td><td>0.12</td><td>217.64</td><td>218.20</td><td>-0.26</td></tr><tr><td>I<sub>rms</sub> (A)</td><td>23.10</td><td>23.21</td><td>-0.48</td><td>4.47</td><td>4.45</td><td>0.45</td></tr><tr><td>Frequency (Hz)</td><td>50</td><td>50</td><td>0.00</td><td>50.00</td><td>50.00</td><td>0.00</td></tr><tr><td>Power Factor</td><td>0.99</td><td>0.99</td><td>0.00</td><td>0.99</td><td>0.99</td><td>0.00</td></tr><tr><td>THD<sub>v</sub> (%)</td><td>1.15</td><td>1.2</td><td>-4.35</td><td>1.18</td><td>1.24</td><td>-5.08</td></tr><tr><td>THD<sub>i</sub> (%)</td><td>3.25</td><td>3.12</td><td>4.00</td><td>3.53</td><td>3.68</td><td>-4.25</td></tr><tr><td>S (VA)</td><td>5044.38</td><td>5060.7</td><td>-0.32</td><td>972.85</td><td>970.99</td><td>0.19</td></tr><tr><td>P (W)</td><td>4993.94</td><td>5010.1</td><td>-0.32</td><td>963.12</td><td>961.28</td><td>0.19</td></tr><tr><td>Q (Var)</td><td>711.59</td><td>713.85</td><td>-0.32</td><td>137.24</td><td>136.97</td><td>0.19</td></tr><tr><td>V p.u.</td><td>0.99</td><td>0.99</td><td>0.00</td><td>0.98</td><td>0.99</td><td>-1.02</td></tr></tbody></table>
+
+Table 4. Comparison of measured and modeled electrical parameters of the FVHC condition and the transient step down condition

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+[6] Muh. Imran Hamid and Makbul Anwari, "Single phase Photovoltaic Inverter Operation Characteristic in Distributed Generation System", Distributed Generation, Intech book, 2010
+
+[7] Vu Van T., Driesen J., Belmans R., Interconnection of distributed generators and their influences on power system. International Energy Journal, vol. 6, no. 1, part 3, pp. 127-140, 2005.
+
+[8] A. Moreno-Munoz, J.J.G. de-la-Rosa, M.A. Lopez-Rodriguez, J.M. Flores-Arias, F.J. Bellido-Outerino, M. Ruiz-de-Adana, "Improvement of power quality using distributed generation", Electrical Power and Energy Systems 32 (2010) 1069-1076
+
+[9] P.R.Khatri, V.S.Jape, N.M.Lokhande, B.S.Motling, "Improving Power Quality by Distributed Generation" Power Engineering Conference, 2005. IPEC 2005. The 7th International.
+
+[10] Vu Van T., Driesen J., Belmans R., Power quality and voltage stability of distribution system with distributed energy resources. International Journal of Distributed Energy Resources, vol. 1, no. 3, pp. 227-240, 2005.
+
+[11] Woyte A., Vu Van T., Belmans R., Nijs J., Voltage fluctuations on distribution level introduced by photovoltaic systems. IEEE Transactions on Energy Conversion, vol. 21, no. 1, pp. 202-209, 2006.
+
+[12] Thongpron, K.Kirtara, Effects of low radiation on the power quality of a distributed PV-grid connected system, Solar Energy Materials and Solar Cells Solar Energy Materials and Solar Cells, Vol. 90, No. 15. (22 September 2006), pp. 2501-2508.
+
+[13] S.K. Khadem, M.Basu and M.F.Conlon, "Power quality in Grid connected Renewable Energy Systems: Role of Custom Power Devices", International conference on Renewable Energies and Power quality (ICREPQ'10), Granada, Spain, 23rd to 25th March, 2010
+
+[14] Mohamed A. Eltawil Zhengming Zhao, "Grid-connected photovoltaic power systems: Technical and potential problems – A review" Renewable and Sustainable Energy Reviews Volume 14, Issue 1, January 2010, Pages 112-129
+
+[15] Soeren Baekhoej Kjaer, et al., "A Review of Single-Phase Grid-Connected inverters for Photovoltaic Modules, EEE Transactions and industry applications, Transactions on Industry Applications, Vol. 41, No. 5, September 2005
+
+[16] F. Blaabjerg, Z. Chen and S. B. Kjaer, "Power Electronics as Efficient Interface in Dispersed Power Generation Systems." IEEE Trans. on Power Electronics 2004; vol.19 no. 5. Pp. 1184-1194. 2000.
+
+[17] Chi, Kong Tse, "Complex behavior of switching power converters", Boca Raton : CRC Press, c2004
+
+[18] Giraud, F., Steady-state performance of a grid-connected rooftop hybrid wind-photovoltaic power system with battery storage, IEEE Transactions on Energy Conversion, 2001.
+
+[19] G.Saccomando, J.Svensson, "Transient Operation of Grid-connected Voltage Source Converter Under Unbalanced Voltage Conditions", IEEE Industry Applications Conference, 2001.
+
+[20] Li Wang and Ying-Hao Lin, "Small-Signal Stability and Transient Analysis of an Autonomous PV System", Transmission and Distribution Conference and Exposition, 2008.

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+in order to check whether they apply to $\mathcal{ALC}_{RA\ominus}$. The discussion provides some key-insights into the high expressiveness of composition-based role axioms.
+
+**$\mathcal{ALC}_{R+}$**: The description logic $\mathcal{ALC}_{R+}$ augments $\mathcal{ALC}$ (see [17]) with transitively closed roles. A role $R$ may be declared as transitively closed which enforces (for every model $\mathcal{I}$) $R^\mathcal{I} = (R^\mathcal{I})^+$. $\mathcal{ALC}_{R+}$ is obviously a proper sub-fragment of $\mathcal{ALC}_{RA\ominus}$, since a role $R$ can be declared as transitively closed with the role axiom $R \circ R \sqsubseteq R$, which enforces $(R^\mathcal{I})^+ \subseteq R^\mathcal{I}$ and therefore $R^\mathcal{I} = (R^\mathcal{I})^+$. The concept satisfiability problem of $\mathcal{ALC}_{R+}$ is decidable and PSPACE-complete. $\mathcal{ALC}_{R+}$ is basically just a syntactic variant of the multi-modal logic $K4_n$, with $n$ transitive accessibility relations; plain $\mathcal{ALC}$ corresponds to $K_n$ (see [18]). The $n$ accessibility relations correspond to $n$ different roles. The only difference between $\mathcal{ALC}_{R+}$ and $K4_n$ is that the latter requires that all $n$ accessibility relations are transitively closed, whereas the transitive closure of a role is optionally in $\mathcal{ALC}_{R+}$.
+
+**$\mathcal{ALC}_+$ and $\mathcal{ALC}_\oplus$:** As Sattler points out, $\mathcal{ALC}_{R+}$ is not capable to distinguish “direct” and “indirect” successors of a transitively closed role $R$. Baader had already introduced the language $\mathcal{ALC}_+$ (see [1]) which provides a transitive closure operator (in fact, $\mathcal{ALC}_+$ is more or less a notational variant of the Propositional Dynamic Logic, PDL, see [18]). Both a “generating” role $R$ and its transitive closure $+(R)$ can be distinguished and used separately within concepts. $(+(R))^\mathcal{I} = (R^\mathcal{I})^+$ is enforced. $\mathcal{ALC}_+$ is no longer a subset of FOPL, since the transitive closure of a role cannot be expressed in FOPL (this violates the compactness of FOPL). However, it can be expressed in FOPL that a role $R$ is transitively closed: $\forall x, y, z : R(x, y) \wedge R(y, z) \Rightarrow R(x, z)$. There is no way to simulate the expressiveness of $\mathcal{ALC}_+$ in $\mathcal{ALC}_{RA\ominus}$, since the latter is still a subset of FOPL, but the former is not. The concept satisfiability problem of $\mathcal{ALC}_+$ is decidable and EXPTIME-complete. In the search for a computationally less expensive logic, Sattler introduced the language $\mathcal{ALC}_\oplus$ (see [17]), which replaces the transitive closure operator “+” with the so-called “transitive orbit” operator “⊕”. Like the “+”-operator, the transitive orbit operator can be applied to roles. Applied to a role $R$, the role $⊕(R)$ is interpreted as some relation being a superset of the transitive closure of the generating role $R$, but not necessarily the smallest one: only $(R^\mathcal{I})^+ \subseteq (⊕(R))^\mathcal{I}$ is granted. The concept satisfiability problem of $\mathcal{ALC}_\oplus$ is decidable but unfortunately, as Sattler has shown, EXPTIME-complete (as for $\mathcal{ALC}_+$, too).
+
+We show that $\mathcal{ALC}_\oplus$ is subsumed by $\mathcal{ALC}_{RA\ominus}$ by reducing the concept satisfiability problem of $\mathcal{ALC}_\oplus$ to the concept satisfiability problem of $\mathcal{ALC}_{RA\ominus}$: given an $\mathcal{ALC}_\oplus$ concept C, we construct a concept C' and a role box $\mathfrak{R}'$ such that

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+The so-called *Guarded Fragment* (*GF*) as introduced by Andréka, van Benthem and Németi is another fragment of FOPL that is decidable (it even has the finite tree model property). We will not formally discuss it here (see [9, 8]). However, its prominent feature is that the number of variables is *not* bounded, as long as certain syntactic restrictions on the use of the quantifiers are obeyed. Grädel suggested to use the guarded fragment as the basis for a new family of *n*-ary DLs (see [8]). Since *FO*³ is undecidable, but *GF*³ (the guarded fragment with three variables) is decidable, decidability for *ALC<sub>RA</sub>*<sup>ø</sup> would follow if *ALC<sub>RA</sub>*<sup>ø</sup> was expressible in *GF*³. A few informal words regarding the guarded fragment seem to be appropriate: when translating propositional modal logics (for example, *ALC* resp. *K<sub>n</sub>* ) into FOPL, one observes that the quantifiers are always used in a certain *guarded way*. The quantifiers appear only in “patterns” of the form ∀x, y : *R(x,y)* ⇒ *C(y)* and ∃x, y : *R(x,y)* ∧ *C(y)*. Here, the atom *R(x,y)* is used as a *guard*. This observation was generalized into the guarded fragment and observed to be responsible for the nice computational properties resp. decidability of many modal logics (and the guarded fragment as well). More specifically, the guard must always be an atom (complex formulas may not be guards) and must contain all variables that appear in the subsequent of the formula “behind” the guard. The formulas ∀x, y : *R(x,y)* ⇒ *C(y)* and ∃x, y : *R(x,y)* ∧ *C(y)* are therefore in *GF*², where *GF*² is the guarded fragment with two variables: *GF*² = *F0*²∩*GF*. If all non-unary atoms are only used as guards, the *GF* formula is said to be *monadic*. This is obviously the case for *ALC*, since the binary relations occur solely as guards. Obviously, the transitivity axiom ∀x,y,z : *R(x,y)* ∧ *R(y,z)* ⇒ *R(x,z)* is *not* in the *GF*. The loosely guarded fragment (*LGF*) is a generalization of the guarded fragment by additionally allowing not only atoms (like *R(x,y*) being guards, but also conjunctions of atoms. However, the transitivity axiom is not even in the *LGF*, since it is additionally required that there must be a conjunct using *x* and *z* in *one guard atom*; e.g. ∀x,y,z : *R(x,y)* ∧ *R(y,z)* ∧ *S(x,z)* ⇒ *R(x,z)* is in the *LGF*, but ∀x,y,z : *R(x,y)* ∧ *R(y,z)* ⇒ *R(x,z)* is not. Grädel has even shown that it is impossible to express that a relation is transitively closed within the guarded or the loosely guarded fragment. The transitivity axiom ∀x,y,z : *R(x,y)* ∧ *R(y,z)* ⇒ *R(x,z)* cannot be expressed by any means (see [9]). But this means that *ALC<sub>RA</sub>*<sup>ø</sup> is not in the *LGF*.
+
+Therefore, the (loosely) guarded fragment has been extended by transitivity
+on an “extra-logical” level (since transitivity is not expressible within the logic
+itself), and the following results have been obtained:³
+
+• $GF^3$ with transitive relations is undecidable (see [9]).
+
+• $\mathit{LGF}_-$ with one transitive relation is undecidable (see [7]).
+
+³A minus suffix indicates that the logic does not provide equality.

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+Figure 3: RCC8 Qualitative Spatial Relationships: *EQ* = Equal, *DC* = Disconnected, *EC* = Externally Connected, *PO* = Partial Overlap, *TPP* = Tangential Proper Part, *NTPP* = Non-Tangential Proper Part. Read the relations as *TPP*(A, B), *NTPP*(A, B) etc. *TPP* and *NTPP* have corresponding inverse relationships: *TPPI* and *NTPPI*, e.g. *TPPI*(B, A), *NTPPI*(B, A).
+
+• Even $GF^2$ with transitive relations is undecidable (see [7]).
+
+• *Monadic* $GF^2$ with binary transitive, symmetric and/or reflexive relations is *decidable* (see [7]).
+
+None of these results is applicable in the case of $\mathcal{ALC}_{RA\ominus}$. The most important result concerning $\mathcal{ALC}_{RA\ominus}$ is the last one, since $\mathcal{ALC}$ is in monadic $GF^2$, and the role box allows one to express, for example, transitivity. However, the role boxes of $\mathcal{ALC}_{RA\ominus}$ can express a lot more than transitivity. Therefore, this result implies the decidability of, e.g. $\mathcal{ALC}_{R+}$, but not of $\mathcal{ALC}_{RA\ominus}$. In fact, a much more general result has been shown by Ganzinger et al. (see [7]), but it does not apply to axioms of the form $\forall x, y, z : R(x, y) \land S(y, z) \Rightarrow T(x, z)$.
+
+# 4 Spatial Reasoning With $\mathcal{ALC}_{RA\ominus}$
+
+A widely accepted approach in the field of spatial reasoning for describing spatial relationships between two-dimensional objects in the plane is to describe their spatial interrelationship qualitatively instead of describing their metrical and/or geometrical attributes. Examples for qualitative spatial calculi fitting into this category are the well-known RCC8 calculus (see [16]) and the so-called Egenhofer-relations (see [6]). In the case of RCC8, we can distinguish 8 disjoint – pairwise exclusive – base relations that describe purely topological aspects of the scene, *exhaustively* covering the space of all possibilities (see Figure 3). Informally speaking this means that between every two objects in the plane *exactly one* of the RCC8 relations holds.
+
+Given a set of base relations, e.g. the RCC8 relations, the most important inference problem is the following: given three regions *a*, *b* and *c* in the plane, and the relations *R*(*a*, *b*), *S*(*b*, *c*) between them, what can be deduced about the possible relationships between *a* and *c*? This basic inference task is usually given by

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+Figure 5: Illustration of a model of *special\_figure*
+
+Then, the question is: does *figure\_touching\_a\_figure* subsume *special\_figure*, or equivalently, is *figure* ⊓ ∀*PO*.¬*figure* ⊓ ∀*NTPPI*.¬*figure* ⊓ ∀*TPPI*.¬*circle* ⊓ ∃*TPPI*.(*figure* ⊓ ∃*EC*.*circle*) ⊓ ¬(*figure* ⊓ ∃*EC*.*figure*) unsatisfiable w.r.t. a role box $\mathfrak{R}$ corresponding to the RCC8 composition table?
+
+After pushing the negation sign inwards and removing the obviously contradictory disjunct from the resulting disjunction, the concept *figure* ⊓ ∀*PO*.¬*figure* ⊓ ∀*NTPPI*.¬*figure* ⊓ ∀*TPPI*.¬*circle* ⊓ ∃*TPPI*.(*figure* ⊓ ∃*EC*.*circle*) ⊓ ∀*EC*.¬*figure* must be unsatisfiable then. Please consider Figure 5 which illustrates a “model” of *special\_figure*, with $a \in special\_figure^I$, $b \in (figure \sqcap \exists EC circle)^I$, and $c \in circle^I$, with $\langle a, b \rangle \in TPPI^I$, $\langle b, c \rangle \in EC^I$; please note that $TPPI \circ EC \subseteq EC \sqcup PO \sqcup TPPI \sqcup NTPPI \in \mathfrak{R}$. Due to the definition of *special\_figure*, it can be seen that in every model $\langle a, c \rangle \in EC^I$ must hold. But then, due to ∀*EC*.¬*figure*, it is obviously the case that *figure* ⊓ ∀*PO*.¬*figure* ⊓ ∀*NTPPI*.¬*figure* ⊓ ∀*TPPI*.¬*circle* ⊓ ∃*TPPI*.(*figure* ⊓ ∃*EC*.*circle*) ⊓ ∀*EC*.¬*figure* has no models and is therefore unsatisfiable. This shows that *special\_figure* is indeed subsumed by *figure\_touching\_a\_figure*.
+
+Please note that there are also other description logics suitable for spatial reasoning tasks, namely the language $\mathcal{ALCRP}(D)$ (see [11]). However, unrestricted $\mathcal{ALCRP}(D)$ is undecidable (see [15]), and its decidable fragment suffers from very strong syntax-restrictions, dramatically pruning the space of allowed concept expressions. In fact, the finite model property is ensured in re-

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+# Ambiguity in the m-bonacci numeration system
+
+Petra Kocábová, Zuzana Masáková, Edita Pelantová
+
+► To cite this version:
+
+Petra Kocábová, Zuzana Masáková, Edita Pelantová. Ambiguity in the m-bonacci numeration system. Discrete Mathematics and Theoretical Computer Science, DMTCS, 2007, 9 (2), pp.109-123. hal-00966537
+
+HAL Id: hal-00966537
+
+https://hal.inria.fr/hal-00966537
+
+Submitted on 26 Mar 2014
+
+**HAL** is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.
+
+L'archive ouverte pluridisciplinaire **HAL**, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

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+# Ambiguity in the *m*-bonacci numeration system
+
+Petra Kocábová and Zuzana Masáková and Edita Pelantová
+
+*Department of Mathematics FNSPE, Czech Technical University*
+*Trojanova 13, 120 00 Praha 2, Czech Republic*
+
+*received 27 Dec 2004, revised 23 Apr 2005, accepted 19 May 2005.*
+
+We study the properties of the function $R^{(m)}(n)$ defined as the number of representations of an integer $n$ as a sum of distinct $m$-Bonacci numbers $F_k^{(m)}$, given by $F_i^{(m)} = 2^{i-1}$, for $i \in \{1, 2, \dots, m\}$, $F_{k+m}^{(m)} = F_{k+m-1}^{(m)} + F_{k+m-2}^{(m)} + \dots + F_k^{(m)}$, for $k \ge 1$. We give a matrix formula for calculating $R^{(m)}(n)$ from the greedy expansion of $n$. We determine the maximum of $R^{(m)}(n)$ for $n$ with greedy expansion of fixed length $k$, i.e. for $F_k^{(m)} \le n < F_{k+1}^{(m)}$. Unlike the Fibonacci case $m=2$, the values of the maxima are not related to the sequence $(F_k^{(m)})_{k \ge 1}$. We describe the palindromic structure of the sequence $(R^{(m)}(n))_{n \in \mathbb{N}}$, which is richer than in the case of Fibonacci numeration system.
+
+**Keywords:** numeration system, generalized Fibonacci numbers, greedy expansion, palindromes
+
+## 1 Introduction
+
+Any strictly increasing sequence $(G_k)_{k \in \mathbb{N}}$, with $G_k \in \mathbb{N}$, $G_1 = 1$, defines a system of numeration where every positive integer can be written as a linear combination $\sum a_k G_k$, where $a_k \in \mathbb{N}_0$, see for instance [8]. Some sequences $(G_k)_{k \in \mathbb{N}}$ have even nicer property: Every positive integer can be expressed as a sum of distinct elements of the sequence $(G_k)_{k \in \mathbb{N}}$. The necessary and sufficient condition so that it is possible is that the sequence satisfies $G_1 = 1$ and $G_n - 1 \le \sum_{i=1}^{n-1} G_i$ for all $n \in \mathbb{N}$. Example of such a sequence is $(2^{k-1})_{k \ge 1}$ or $(F_k)_{k \ge 1}$, the sequence of Fibonacci numbers. The expression of $n$ in the form
+
+$$n = G_{i_s} + G_{i_{s-1}} + \cdots + G_{i_1}, \quad \text{where } i_s > i_{s-1} > \cdots > i_1 \ge 1,$$
+
+is called a representation of $n$ in the numeration system $(G_k)_{k \in \mathbb{N}}$. This representation can be written using a sequence of coefficients $(a_k)_{k \in \mathbb{N}} \in \{0, 1\}^\mathbb{N}$ as $n = \sum_{i=1}^\infty a_i G_i$, where only a finite number of elements of the sequence $(a_k)_{k \in \mathbb{N}}$ are non-zero. The maximal index $k$ such that $a_k \neq 0$ are called the length of the representation. The representation can be coded by the word $a_k a_{k-1} \cdots a_1$ in the alphabet $\{0, 1\}$. In the writing of number representations we adopt the usual convention that the concatenation of $l$ copies of a finite word is written $w^l$, for $l = 0, 1, 2, \dots$. For example, the representation of the number $n = G_{k+2} + G_k$ is coded by the word $1010^{k-1}$.
+
+If $(G_k)_{k \in \mathbb{N}} = (2^{k-1})_{k \ge 1}$, then the representation of every integer $n$ in the system $(G_k)_{k \in \mathbb{N}}$ is unique, and the word $a_k a_{k-1} \cdots a_1$ is the binary expansion of $n$. If we choose for $(G_k)_{k \in \mathbb{N}}$ the

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+Fibonacci sequence $G_k = F_k$, given by the recurrence $F_{k+2} = F_{k+1} + F_k$, $F_0 = F_1 = 1$, then most integers have several representations. The number of distinct representations, denoted by $R(n)$, is a function studied by many authors [1, 2, 5, 11].
+
+On the set of representations of a given integer $n$ in the system $(G_k)_{k \in \mathbb{N}}$ one can introduce the lexicographic order in the following way: We say that the representation $v_l v_{l-1} \cdots v_1$ of the number $n$ is greater than the representation $u_k u_{k-1} \cdots u_1$ of $n$ if $l > k$ or $l = k$ and the first non-zero element in the sequence $v_k - u_k, v_{k-1} - u_{k-1}, \cdots, v_1 - u_1$ is positive. This order is sometimes called the radix order. The lexicographically greatest representation of a given number $n$ is called the greedy expansion of $n$.
+
+In this paper we study the measure of ambiguity of the representation of integers in the generalized Fibonacci numeration systems, the so-called *m*-Bonacci systems defined for $m \ge 2$ by recurrence
+
+$$ 
+\begin{aligned}
+& F_1^{(m)} = 1, \quad F_2^{(m)} = 2, \quad \dots, \quad F_m^{(m)} = 2^{m-1}, \\
+& F_{k+m}^{(m)} = F_{k+m-1}^{(m)} + F_{k+m-2}^{(m)} + \dots + F_k^{(m)}, \quad \text{for } k \ge 1.
+\end{aligned} 
+\tag{1} $$
+
+The 2-Bonacci sequence is thus the ordinary Fibonacci sequence; 3-Bonacci sequence is usually called the Tribonacci sequence. Combinatorial properties of the *m*-Bonacci numeration system have been discussed in [7], in order to study the Garsia entropy connected with Pisot numbers $\beta$ fulfilling $\beta^m = \beta^{m-1} + \cdots + \beta + 1$.
+
+The *m*-Bonacci numeration systems are studied in [6] from the point of view of automata theory. It is proven that addition of integers written in the *m*-Bonacci numeration system can be performed by means of a finite state automaton, whereas it is impossible to convert an *m*-Bonacci representation of an integer into its standard binary expansion by a finite state automaton.
+
+It has been shown already in [10] that every non-negative integer $n$ can be represented as a sum of distinct elements of the *m*-Bonacci sequence. Such representation of $n$ may not be unique. We denote by $R^{(m)}(n)$ the number of different representations of $n$. The recurrence relation for *m*-Bonacci numbers ensures that starting from an arbitrary representation of $n$ we can get any other representation of $n$ by interchanging $10^m \leftrightarrow 01^m$ or vice versa in the word coding the representation of $n$.
+
+Obviously, the lexicographically greatest (greedy) representation of $n$ does not contain the block $1^m$. Let us denote the greedy expansion of $n$ in the numeration system $(F_k^{(m)})_{k \ge 1}$ by $\langle n \rangle_m$. It can be written in the form
+
+$$ \langle n \rangle_m = 10^{r_s} 10^{r_{s-1}} 10^{r_{s-2}} \cdots 10^{r_2} 10^r, \quad \text{where } r_i \in \mathbb{N}_0, $$
+
+and for every $i$ such that $m-1 \le i \le s$ we have $r_{i-m+2} + r_{i-m+3} + \cdots + r_i \ge 1$. The length of the greedy expansion $\langle n \rangle_m$ is $s+r_s+r_{s-1}+\cdots+r_1$. If the lengths of $\langle n \rangle_m$ is $k$, then every other representation of $n$ has the length either $k$ or $k-1$. Representations of length $k$ are called ‘long’ representations and their number is denoted by $R_1^{(m)}(n)$; the other representations are called ‘short’ and their number is denoted by $R_0^{(m)}(n)$. Obviously, we have
+
+$$ R^{(m)}(n) = R_0^{(m)}(n) + R_1^{(m)}(n). $$
+
+The aim of the paper is to study the properties of the function $R^{(m)}(n)$. First we show that the Berstel matrix formula [1] for calculation of the value $R^{(2)}(n)$ from the greedy expansion $\langle n \rangle_m$ can

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+be generalized for $m \ge 3$. In the next section we focus on the study of the segment of the sequence $R^{(m)}(n)$ for $F_k^{(m)} \le n < F_{k+1}^{(m)}$, i.e. for such integers $n$ whose greedy expansion has constant length $k$. For the Fibonacci numeration system it is known [2, 5] that among numbers with a fixed length $k$ of the greedy expansion only $n = F_{k+1}^{(2)} - 1$ satisfies $R^{(2)}(n) = 1$, and moreover, the segments of the sequence $R^{(2)}(n)$ between two unit values are palindromes. For the $m$-Bonacci numeration system with $m \ge 3$ we show that the number of integers $n$ in the segment $[F_k^{(m)}, F_{k+1}^{(m)})$ with a unique representation $R^{(m)}(n) = 1$ is equal to the $(m-1)$-Bonacci number $F_k^{(m-1)}$. Thus the number of 1's in the corresponding segment of the sequence $(R^{(m)}(n))_{n \in \mathbb{N}}$ increases, however, we show that the palindromic structure of the sequence $(R^{(m)}(n))_{n \in \mathbb{N}}$ remains preserved.
+
+In the rest of the paper we determine the maximum of the function $(R^{(m)}(n))_{n \in \mathbb{N}}$ in the mentioned segment. For $m=2$, i.e. the Fibonacci numeration system, the maxima have been determined in [11],
+
+$$ \max\{R^{(2)}(n) \mid F_k^{(2)} \le n < F_{k+1}^{(2)}\} = \begin{cases} F_{\frac{k+1}{2}}^{(2)} & \text{for } k \text{ odd,} \\ 2F_{\frac{k-2}{2}}^{(2)} & \text{for } k \text{ even.} \end{cases} $$
+
+We shall thus concentrate on determining the values of the maxima for $m \ge 3$. Unlike the Fibonacci case, the values of the maxima are not related to the sequence $(F_k^{(m)})_{k \ge 1}$.
+
+## 2 The number of representations of $n$ in the $m$-Bonacci system
+
+The number of representations of a given integer $n$ is related to the possible interchanges $10^m \leftrightarrow 01^m$ in the greedy expansion of $n$. For example, if $\langle n \rangle_m$ is of length $k \le m$, then no interchange is possible and we have $R^{(m)}(n) = 1$. If the length of $\langle n \rangle_m$ is $m+1$, then only $\langle n \rangle_m = 10^m$ admits such an interchange. It follows that
+
+$$ R^{(m)}(n) = 1, \quad \text{for } 1 \le n \le F_{m+2}^{(m)} - 1, \quad n \ne F_{m+1}^{(m)}, \\ R^{(m)}(F_{m+1}^{(m)}) = 2. \tag{2} $$
+
+The aim of this section is to derive a compact formula for calculating the values of the function $R^{(m)}(n)$. Both the formula and its proof are slight generalizations of the result of [1, 5] for the case $m=2$. Consider therefore $m \ge 3$.
+
+First we state several simple observations, which transpose the calculation of the value $R_0^{(m)}(n)$ and $R_1^{(m)}(n)$ for an integer $n$ with $s$ 1's in its greedy expansion to calculation of $R_0^{(m)}(n)$ and $R_1^{(m)}(n)$ for some $\tilde{n}$ whose greedy expansion has strictly smaller number $\tilde{s} < s$ of 1's. In the following, we shall identify the writing $R^{(m)}(n)$ with $R^{(m)}(\tilde{n})$, where $\tilde{n}$ is the word in the alphabet $\{0,1\}$ coding the greedy expansion of $n$, i.e. a word starting with 1.
+
+**Fact 2.1** If $0 \le l \le m-2$, then $R_0^{(m)}(10^l w) = 0$, therefore $R_1^{(m)}(10^l w) = R^{(m)}(\tilde{w})$. In matrix form,
+
+$$ \begin{pmatrix} R_0^{(m)}(10^l w) \\ R_1^{(m)}(10^l w) \end{pmatrix} = \begin{pmatrix} 0 & 0 \\ 1 & 1 \end{pmatrix} \begin{pmatrix} R_0^{(m)}(\tilde{w}) \\ R_1^{(m)}(\tilde{w}) \end{pmatrix}. $$

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+**Remark 2.2**
+
+(i) Note that $R_0^{(m)}(10^l w) = 0$ does not imply $l \le m-2$. It is not difficult to see that $R_0^{(m)}(w) = 0$
+implies that the word $w$ is of the form $w = (10^{m-1})^s \tilde{w}$, where $s \ge 0$ and $\tilde{w}$ is either the
+empty word, or a word of the form $\tilde{w} = 10^l \tilde{w}$ for $l \le m-2$. Therefore the smallest $n_1$ such
+that $\langle n_1 \rangle_m = k$ and for which $R_0^{(m)}(n_1) = 0$ is the number with greedy expansion
+
+$$
+\langle n_1 \rangle_m = \begin{cases} (10^{m-1})^s, & \text{if } s := \frac{k}{m} \in \mathbb{N}, \\ (10^{m-1})^s 10^{k-ms-1}, & \text{if } s := \lfloor \frac{k}{m} \rfloor \neq \frac{k}{m}. \end{cases} \tag{3}
+$$
+
+At the same time, for every $n$ such that $n_1 \le n < F_{k+1}^{(m)}$ we have $R_0^{(m)}(n) = 0$.
+
+(ii) In the word $\langle n_1 \rangle_m$ of the form (3) one cannot perform any interchange $10^m \leftrightarrow 01^m$, and therefore $R^{(m)}(n_1) = 1$. We have thus found the smallest number $n$ such that $F_k^{(m)} < n < F_{k+1}^{(m)}$ and $R^{(m)}(n) = 1$. Note that in the Fibonacci numeration system $R_0^{(2)}(n) = 0$ already implies $R^{(2)}(n) = 1$. For $m \ge 3$ this is not valid. As an example, consider $\langle n \rangle_m = 110^{k-2}$ for $k \ge m+2$. Such $n$ satisfies $R_0^{(m)}(n) = 0$ and $R^{(m)}(n) \ge 2$.
+
+(iii) Let us express explicitly the value of $n_1$. Every 1 in the word $\langle n_1 \rangle_m$ at the position $i > m$ represents the number
+
+$$
+F_i^{(m)} = F_{i-1}^{(m)} + F_{i-2}^{(m)} + \dots + F_{i-m}^{(m)}. \\
+\text{The 1 at a position } i \le m \text{ represents}
+$$
+
+$$
+F_i^{(m)} = 2^{i-1} = 1 + F_{i-1}^{(m)} + F_{i-2}^{(m)} + \cdots + F_1^{(m)}.
+$$
+
+The number $n_1$ with greedy expansion of the form (3) is therefore equal to
+
+$$
+n_1 = 1 + \sum_{i=1}^{k-1} F_i^{(m)} .
+$$
+
+**Fact 2.3** $R_0^{(m)}(10^{m-1}w) = R_0^{(m)}(w)$ and $R_1^{(m)}(10^{m-1}w) = R^{(m)}(w)$. In a matrix form,
+
+$$
+\begin{pmatrix} R_0^{(m)}(10^{m-1}w) \\ R_1^{(m)}(10^{m-1}w) \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix} \begin{pmatrix} R_0^{(m)}(w) \\ R_1^{(m)}(w) \end{pmatrix}.
+$$
+
+**Fact 2.4** If $l \ge m$, then we have $R_0^{(m)}(10^l w) = R^{(m)}(10^{l-m} w)$ and $R_1^{(m)}(10^l w) = R_1^{(m)}(10^{l-m} w)$.
+In a matrix form,
+
+$$
+\begin{pmatrix} R_0^{(m)}(10^l w) \\ R_1^{(m)}(10^l w) \end{pmatrix} = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix} \begin{pmatrix} R_0^{(m)}(10^{l-m} w) \\ R_1^{(m)}(10^{l-m} w) \end{pmatrix}.
+$$
+
+**Lemma 2.5** Let $\langle n \rangle_m = 10^l w$, where $w = (\tilde{n})_m$ for some integer $\tilde{n}$. Then
+
+$$
+\left(\begin{array}{c}
+R_{0}^{(m)}(10^{l} w) \\
+R_{1}^{(m)}(10^{l} w)
+\end{array}\right)=\left(\begin{array}{cc}
+\left[\frac{l+1}{m}\right] & \left[\frac{l}{m}\right] \\
+1 & 1
+\end{array}\right)\left(\begin{array}{c}
+R_{0}^{(m)}(w) \\
+R_{1}^{(m)}(w)
+\end{array}\right).
+$$

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+**Proof:** Let us write $l = am+b$, where $b \in \{0, 1, \dots, m-1\}$. If $b \le m-2$, then for the calculation of the values $R_0^{(m)}(10^l w)$, $R_1^{(m)}(10^l w)$ one uses $a$ times Fact 2.4 and then Fact 2.1. Since
+
+$$ \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}^a \begin{pmatrix} 0 & 0 \\ 1 & 1 \end{pmatrix} = \begin{pmatrix} 1 & a \\ 0 & 1 \end{pmatrix} \begin{pmatrix} 0 & 0 \\ 1 & 1 \end{pmatrix} = \begin{pmatrix} a & a \\ 1 & 1 \end{pmatrix} = \begin{pmatrix} \left[\frac{l+1}{m}\right] & \left[\frac{l}{m}\right] \\ 1 & 1 \end{pmatrix}, $$
+
+the statement is proved.
+
+If $b=m-1$, we use $a$ times Fact 2.4 and then Fact 2.3. The matrix identity
+
+$$ \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}^a \begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix} = \begin{pmatrix} a+1 & a \\ 1 & 1 \end{pmatrix} = \begin{pmatrix} \left[\frac{l+1}{m}\right] & \left[\frac{l}{m}\right] \\ 1 & 1 \end{pmatrix}, $$
+
+completes the proof. $\square$
+
+In order to derive the formula for calculation of $R^{(m)}(n)$, we need to derive the values $R_0^{(m)}(n)$, $R_1^{(m)}(n)$ for integers $n$ with only one 1 in their greedy expansion. It is easy to see that $R_1^{(m)}(10^l) = 1$ and $R_0^{(m)}(10^l) = [\frac{l}{m}]$, which can be written by
+
+$$ \begin{pmatrix} R_0^{(m)}(10^l) \\ R_1^{(m)}(10^l) \end{pmatrix} = \begin{pmatrix} \left[ \frac{l+1}{m} \right] & \left[ \frac{l}{m} \right] \\ 1 & 1 \end{pmatrix} \begin{pmatrix} 0 \\ 1 \end{pmatrix}. $$
+
+Since $R^{(m)}(n) = R_0^{(m)}(n) + R_1^{(m)}(n)$, we can formulate the result. For that we introduce the following notation,
+
+$$ M(l) = M_m(l) := \begin{pmatrix} \left\lfloor \frac{l+1}{m} \right\rfloor & \left\lfloor \frac{l}{m} \right\rfloor \\ 1 & 1 \end{pmatrix}. \qquad (4) $$
+
+**Theorem 2.6** Let $\langle n \rangle_m = 10^{r_s} 10^{r_{s-1}} \cdots 10^{r_1}$ be the greedy expansion of the integer $n$ in the *m*-Bonacci numeration system. Then
+
+$$ R^{(m)}(n) = (1\ 1) M(r_s) M(r_{s-1}) \cdots M(r_1) \binom{0}{1}. \qquad (5) $$
+
+### 3 Integers with a unique representation in the *m*-Bonacci numeration system
+
+In order that an integer $n$ has only one representation in the *m*-Bonacci numeration system, the lexicographically greatest and the lexicographically smallest representation must coincide. Consider $n$ in the interval $[F_k^{(m)}, F_{k+1}^{(m)})$. The word coding the greedy expansion of $n$ has the form
+
+$$ u_k u_{k-1} \cdots u_1, \quad \text{where } u_1, \ldots, u_{k-1} \in \{0, 1\} \text{ and } u_k = 1. \qquad (6) $$
+
+If $k < m$, then an arbitrary word of the above form is a greedy expansion of some integer $n$. At the same time it is obvious that in such a word one cannot perform any interchange $10^m \leftrightarrow 01^m$ and therefore this integer $n$ has only one *m*-Bonacci representation. Let
+
+$$ U_k^{(m)} = \#\{n \mid F_k^{(m)} \le n < F_{k+1}^{(m)}, R^{(m)}(n) = 1\}. $$

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+We have derived that
+
+$$U_k^{(m)} = 2^{k-1}, \quad \text{for } k = 1, 2, \dots, m-1. \tag{7}$$
+
+Consider now $k \ge m$. A word of length $k$ satisfying (6) is a greedy expansion of some integer $n$, if and only if it does not contain the string $1^m$. In order that no interchange $10^m \leftrightarrow 01^m$ is possible in this word, so that $R^{(m)}(n) = 1$, the word cannot contain the string $0^m$. Therefore $U_k^{(m)}$ is equal to the number of words $u_k u_{k-1} \cdots u_1$ such that
+
+$$\begin{align}
+& u_1, \dots, u_{k-1} \in \{0,1\}, \quad u_k = 1, \quad \text{and} \nonumber \\
+& u_k u_{k-1} \cdots u_1 \text{ does not contain the strings } 0^m, 1^m. \tag{8}
+\end{align}$$
+
+In order to determine $U_k^{(m)}$, we divide words satisfying (8) into $2(m-1)$ disjoint groups according to their suffix
+
+$$v \in S := \{10, 10^2, 10^3, \ldots, 10^{m-1}, 01, 01^2, 01^3, \ldots, 01^{m-1}\}.$$
+
+The number of words satisfying (8) with suffix $v$ will be denoted by $A_k^v$. Obviously,
+
+$$U_k^{(m)} = \sum_{v \in S} A_k^v.$$
+
+Since every word $w$ of length $k$ satisfying (8) is of the form $w = \tilde{w}0$ or $w = \tilde{w}1$, where $\tilde{w}$ is a word of length $k-1$ satisfying (8), we obtain recurrence relations
+
+$$A_k^{10} = A_{k-1}^{01} + A_{k-1}^{012} + \cdots + A_{k-1}^{01^{m-1}}, \tag{9}$$
+
+$$A_k^{10l} = A_{k-1}^{10l-1}, \quad \text{for } l = 2, 3, \ldots, m-1, \tag{10}$$
+
+$$A_k^{01} = A_{k-1}^{10} + A_{k-1}^{102} + \cdots + A_{k-1}^{10^{m-1}}, \tag{11}$$
+
+$$A_k^{01l} = A_{k-1}^{01l-1}, \quad \text{for } l = 2, 3, \ldots, m-1. \tag{12}$$
+
+Equations (9) and (11) imply that $U_k^{(m)} = A_{k+1}^{10} + A_{k+1}^{01}$. From (10) we obtain $A_k^{10l} = A_{k-l+1}^{10}$ for $l=2,3,\dots,m-1$. Similarly, from (12) we obtain $A_k^{01l} = A_{k-l+1}^{01}$ for $l=2,3,\dots,m-1$. Substituting this into (9) and (11) and taking sum, we obtain
+
+$$U_k^{(m)} = A_{k+1}^{10} + A_{k+1}^{01} = (A_k^{10} + A_k^{01}) + (A_{k-1}^{10} + A_{k-1}^{01}) + \cdots + (A_{k-m+2}^{10} + A_{k-m+2}^{01}).$$
+
+The sequence $(A_{k+1}^{10} + A_{k+1}^{01})_{k \in \mathbb{N}} = (U_k^{(m)})_{k \in \mathbb{N}}$ thus satisfies the same recurrence relation as the $(m-1)$-Bonacci sequence $F_k^{(m-1)}$. It has even the same initial conditions (cf. (1) and (7)). We have thus derived the following statement.
+
+**Proposition 3.1** For $m \ge 3$ the number of integers $n$ with greedy expansion of length $k$ having unique representation in the $m$-Bonacci numeration system is equal to the $k$-th element of the $(m-1)$-Bonacci system. Formally,
+
+$$\#\{n \mid F_k^{(m)} \le n < F_{k+1}^{(m)} \text{ and } R^{(m)}(n) = 1\} = F_k^{(m-1)}.$$
+
+A general theory for counting the number of words with forbidden strings is developed in [9].

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+# 4 Palindromic structure of $R^{(m)}(n)$
+
+Let us recall that transition between different representations of the same integer $n$ is allowed by the interchange $10^m \leftrightarrow 01^m$. Note that the block $10^m$ is the complement of the block $01^m$, in the sense that every 1 is substituted by 0 and every 0 is substituted by 1. Taking complement of the word $\langle n \rangle_m = 1u_{k-1} \cdots u_1$, we obtain the word $(1-u_{k-1})(1-u_{k-2})\cdots(1-u_1)$, which is an $m$-Bonacci representation of an integer, which we denote by $\bar{n}$. It is obvious that
+
+$$R^{(m)}(n) = R^{(m)}(\bar{n}).$$
+
+Since
+
+$$n + \bar{n} = \sum_{i=1}^{k} F_{i}^{(m)} \qquad (13)$$
+
+the center of the symmetry of the function $R^{(m)}(n)$ is in the value $c = \frac{1}{2} \sum_{i=1}^{k} F_i^{(m)}$. Thus the sequence $(R^{(m)}(n))_{n \in \mathbb{N}}$ contains a palindrome, which ends with the value $R^{(m)}(F_{k+1}^{(m)} - 1)$ and starts with the value $R^{(m)}(\sum_{i=1}^{k} F_i^{(m)} - F_{k+1}^{(m)} + 1)$. Note that the center of the symmetry $c$ satisfies $F_k^{(m)} < c < F_{k+1}^{(m)}$ for $k \ge m + 2$. According to (2), the values $R^{(m)}(1), \dots, R^{(m)}(F_{m+2}^{(m)} - 1)$ are all equal to 1 except $R^{(m)}(F_{m+1}^{(m)}) = 2$, thus only $k \ge m + 2$ is interesting.
+
+**Remark 4.1** For $m=2$, i.e. for the Fibonacci sequence, we have $\sum_{i=1}^k F_i^{(2)} = F_{k+2}^{(2)} - 2$. Thus the beginning of the palindrome is at $F_k^{(2)} - 1$ and the end at $F_{k+1}^{(2)} - 1$.
+
+**Remark 4.2** For $m \ge 3$, we have for the starting index of the palindrome
+
+$$\sum_{i=1}^{k} F_{i}^{(m)} - F_{k+1}^{(m)} + 1 < F_{k-1}^{(m)}.$$
+
+Therefore having calculated the values of the function $R^{(m)}(n)$ for $n \le F_k^{(m)} - 1$, most of the values $R^{(m)}(n)$ for $F_k^{(m)} \le n < F_{k+1}^{(m)}$ can be obtained from the palindromic structure.
+
+Let us determine the smallest number $n_0 \in [F_k^{(m)}, F_{k+1}^{(m)})$, whose complement $\bar{n}_0$ lies in the range $[1, F_k^{(m)})$, where we assume having the knowledge of the values of $R^{(m)}$. Obviously $\bar{n}_0 = F_k^{(m)} - 1$ and from (13) we have $n_0 = \sum_{i=1}^{k-1} F_i^{(m)} + 1$. For $k \ge m+2$ we have $n_0 > F_k$. Note that $n_0$ is the same as the number $n_1$ from Remark 2.2. Thus the values $R^{(m)}(n)$ for $\bar{n}_0+1 = F_k^{(m)} \le n \le n_0-1$ are not equal to 1. The sequence $R^{(m)}(\bar{n}_0+1), R^{(m)}(\bar{n}_0+2), \dots, R^{(m)}(n_0-1)$ is a palindrome which does not contain the number 1.
+
+**Example 4.3** For the Tribonacci numeration system, i.e. for $m=3$, the values of the function $R_3^{(m)}$ between $F_7^{(3)} = 44$ and $F_8^{(3)} - 1 = 80$ are the following.
+
+$$ 
+\begin{array}{ccc}
+&R^{(3)}(44) & n_0 = 52 \\
+&\downarrow & \\
+&322222231111122111112111222211121111 \\
+&\uparrow & \\
+\text{center of the palindrome}
+\end{array}
+\qquad
+\begin{array}{ccc}
+&R^{(3)}(80) \\
+&\downarrow \\
+& \\
+\downarrow
+\end{array}
+$$

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+Note that the value 1 appears in the line 21 times, where $21 = F_7^{(2)}$ as corresponds to Proposition 3.1. The line does not show the entire palindrome; the missing values are $R^{(3)}(15), \dots, R^{(3)}(43)$.
+
+We end this section with a theorem whose proof for $m = 2$ can be found in [2, 5]. The proof for $m \ge 3$ follows by induction on the length of the greedy expansion of $n$ from Remark 4.2.
+
+**Theorem 4.4** The segment of the sequence $R^{(m)}(n)$ between two consecutive 1's forms a palindrome, i.e. if $R^{(m)}(p) = R^{(m)}(q) = 1$ and $R^{(m)}(n) > 1$ for all $n, p < n < q$, then the sequence $R^{(m)}(p), R^{(m)}(p+1), \dots, R^{(m)}(q-1), R^{(m)}(q)$ is invariant under mirror image.
+
+## 5 Maxima of the function $R^{(m)}(n)$
+
+The aim of this section is to determine the maximal value of the function $R^{(m)}$ on integers with a fixed length of the greedy expansion. Denote
+
+$$ \text{Max}(k) := \max\{R^{(m)}(n) \mid F_k^{(m)} \le n < F_{k+1}^{(m)}\}. $$
+
+The values $\text{Max}(k)$ for small $k$ can be determined easily. We will use them as the initial step for the proof of the main theorem, which will be done by induction.
+
+* If $k \le m$, then in the expansion of the length $k$ one cannot perform any interchange $10^m \leftrightarrow 01^m$. Thus
+
+$$ \text{Max}(k) = 1, \quad \text{for } 1 \le k \le m. $$
+
+* For $m < k \le 2m$ one can perform at most one interchange $10^m \leftrightarrow 01^m$ and therefore
+
+$$ \text{Max}(k) = 2, \quad \text{for } m + 1 \le k \le 2m. $$
+
+* For $k = 2m+1$ one can perform in the strings $10^{m-1}10^m$ and $10^{2m}$ two interchanges in a given order. Therefore
+
+$$ \text{Max}(2m+1) = 3. $$
+
+* For $2m+1 < k \le 3m$ one can perform on suitable chosen words two independent interchanges. Therefore
+
+$$ \text{Max}(k) = 4, \quad \text{for } 2m+2 \le k \le 3m. $$
+
+* For $k = 3m+1$ we can see by similar arguments that
+
+$$ \text{Max}(3m+1) = 5. $$
+
+In order to obtain a lower bound on $\text{Max}(k)$, we determine the value $R^{(m)}(n)$ on the integers represented by the following words: for $k = a(m+1), a(m+1)+1, \dots, a(m+1)+m-2$ consider $n$ with the greedy expansion of the form
+
+$$ \langle n \rangle_m = (10^m)^a, \ 1(10^m)^a, \ 10(10^m)^a, \ \dots, 10^{m-2}(10^m)^a. $$

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+**Proof:** The string $(10^m)^2 10^{2m-3}$ has the same length as $10^{m-1} 10^{3m-1}$ and the corresponding matrix, $M^2(m)M(2m-3) = \binom{4}{4}$, majores the matrix $M(m-1)M(3m-1) = \binom{3}{4}$ corresponding to the string $10^{m-1} 10^{3m-1}$. $\square$
+
+**Claim 5.6** *The string $10^{m-1}10^{m-1}$ is forbidden for maximality.*
+
+**Proof:** The string $10^{2m-1}$ has the same length as $10^{m-1}10^{m-1}$ and the corresponding matrix $M(2m-1) = \binom{2}{1}$ majores the matrix $M^2(m-1) = \binom{1}{2}$ corresponding to the string $10^{m-1}10^{m-1}$. $\square$
+
+**Claim 5.7** *The string $10^{m-1}10^{2m-1}10^{m-1}$ is forbidden for maximality.*
+
+**Proof:** The string $(10^m)^2 10^{2m-3}$ has the same length and the corresponding matrix $M^2(m)M(2m-3) = \binom{4}{4}$ majores the matrix $M(m-1)M(2m-1)M(m-1) = \binom{3}{5}$ corresponding to the string $10^{m-1}10^{2m-1}10^{m-1}$. $\square$
+
+**Claim 5.8** *The string $10^{m-1}10^{2m-1}10^{2m-1}$ is forbidden for maximality for $m \ge 4$.*
+
+**Proof:** The string $10^{m-1}10^{2m-1}10^{2m-1}$ has the length $5m$ and the corresponding matrix is $M(m-1)M(2m-1)M(m-1) = \binom{5}{8}$. Such matrix is majored by the matrix $M^3(m)M(2m-4) = \binom{8}{8}$ corresponding to the string $(10^m)^3 10^{2m-4}$, which has the same length $5m$. $\square$
+
+**Claim 5.9** *The string $10^{m-1}10^{2m-1}10^{3m-1}$ is forbidden for maximality.*
+
+**Proof:** The string $10^{m-1}10^{2m-1}10^{3m-1}$ with the corresponding matrix is $M(m-1)M(2m-1)M(3m-1) = \binom{7}{11}\binom{5}{8}$. has the length as the string $10^{2m} 10^{2m} 10^{2m-3}$ whose matrix is $M^2(2m)M(2m-3) = \binom{12}{6}\binom{12}{6}$. $\square$
+
+**Remark 5.10** In searching the maximal values $\text{Max}(k)$ for $k \ge m+1$ one can restrict the consideration to integers $n$ such that in their greedy expansion $\langle n \rangle_m = 10^{r_s} \cdots 10^{r_1}$ the coefficient $r_1$ satisfies $r_1 = m$ or $r_1 = 2m$. For, $r_1 \le m-1$ implies that $R^{(m)}(10^{r_s} 10^{r_{s-1}} \cdots 10^{r_3} 10^{r_2+r_1+1}) \ge R^{(m)}(n)$, as follows from the matrix formula. Similarly, $r_1 = am+b$ with $a \ge 1$, $b \in \{1, \dots, m-1\}$, implies that $R^{(m)}(10^{r_s} 10^{r_{s-1}} \cdots 10^{r_2+b} 10^{am}) \ge R^{(m)}(n)$. Claim 5.4 moreover implies that $r_1 \in \{m, 2m\}$.
+
+**Theorem 5.11** Let $m, a$ be integers $m \ge 3$, $a \ge 1$. Then
+
+$$ 
+\begin{align*}
+\operatorname{Max}(a(m+1)+b) &= 2^a && \text{for } b \in \{0, 1, \dots, m-2\}, \\
+\operatorname{Max}(a(m+1)+b-1) &= 2^a + 2^{a-2} && \text{if } a \ge 2, \\
+\operatorname{Max}(a(m+1)+b) &= 2^a + 2^{a-1}. && 
+\end{align*}
+$$

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+**Proof:** The proof is done by induction on the length $k = a(m+1)+b$ of the greedy expansion. The veracity of the statement for the initial values has been established at the beginning of this section. We have also proved that the maxima are greater or equal to the mentioned values. It is therefore sufficient to show that these values are also upper bounds on the maxima.
+
+Assume that $n$ is the argument of the maximum $\text{Max}(k)$. We show that the structure of strings of 0's in the greedy expansion $\langle n \rangle_m = 10^{r_s} 10^{r_{s-1}} \cdots 10^{r_1}$ is only of certain form. First suppose that $\text{R}_0^{(m)}(n) = 0$. Then
+
+$$\text{Max}(k) = \text{R}^{(m)}(10^{r_s} 10^{r_{s-1}} \cdots 10^{r_1}) = \text{R}^{(m)}(10^{r_{s-1}} \cdots 10^{r_1}) \le \text{Max}(k - r_s - 1),$$
+
+and the statement follows from the induction hypothesis. It is therefore sufficient to consider $n$ such that $\text{R}_0^{(m)}(n) \ge 1$. According to Remark 2.2, the greedy expansion $\langle n \rangle_m$ of $n$ is lexicographically smaller than $\langle n_1 \rangle_m$. Together with Claim 5.6 it implies that
+
+$$\langle n \rangle_m = (10^{m-1})^x 10^y w, \quad \text{where } x \in \{0, 1\}, y \ge m, \tag{16}$$
+
+and $w$ is the empty word or the greedy expansion of an integer.
+
+We show that the coefficients $r_i$ (and in particular the exponent $y$) can take only certain values. If there exists an index $i$ such that $0 \le r_i \le m-2$, then (16) implies $i < s$. Since $M(r_i) = \binom{0}{1} \binom{0}{1}$, we have $M(r_{i+1})M(r_i) = \left(\binom{r_{i+1}}{1} \binom{r_{i+1}}{1}\right)$. This implies
+
+$$\text{Max}(k) = R^{(m)}(10^{r_s} \cdots 10^{r_1}) \le R^{(m)}(10^{r_s} \cdots 10^{r_{i+1}} 10^{r_{i-1}} \cdots 10^{r_1}) \le \text{Max}(k - r_i - 1),$$
+
+and the statement follows from the induction hypothesis.
+
+Similarly, if there exists $i$ such that $m+1 \le r_i \le 2m-2$, then $M(r_i) = M(m)$, and therefore $\text{Max}(k) \le \text{Max}(k-r_i+m)$, and again the statement follows from the induction hypothesis. Therefore using Claim 5.4 and Remark 5.10 we can restrict our consideration to coefficients $r_s, r_{s-1}, \dots, r_2 \in \{m-1, m, 2m-1, 2m, 3m-1\}$ and $r_1 \in \{m, 2m\}$.
+
+It follows that $y$ in (16) takes only values $y \in \{m, 2m-1, 2m, 3m-1\}$. We shall now discuss the possibilities according to the values of $x$ and $y$.
+
+$x=1$: Let us discuss the case $x=1$. The condition $y \ge m$ and Claim 5.5 say that $y \in \{m, 2m, 2m-1\}$.
+
+• Let $\langle n \rangle_m = 10^{m-1} 10^m w$, where the length of the word $w$ is $k-(2m+1)$. Since
+
+$$(1\ 1)M(r_s)M(r_{s-1}) = (1\ 1)\binom{1\ 0}{1\ 1}\binom{1\ 1}{1\ 1} = 3(1\ 1),$$
+
+we have $\text{Max}(k) = 3\text{R}^{(m)}(w) \le 3\text{Max}(k-2m-1)$. We express $k = a(m+1)+b$, where $b \in \{0, 1, \dots, m\}$. Therefore
+
+$$\text{Max}(a(m+1)+b) \le 3\text{Max}((a-2)(m+1)+b+1).$$
+
+We distinguish:

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+- If $b \in \{0, 1, \dots, m-2\}$, then using the induction hypothesis
+
+$$ \text{Max}(a(m+1)+b) \leq 3(2^{a-2} + 2^{a-4}) < 2^a, $$
+
+as required.
+
+- If $b=m-1$, then similarly $\text{Max}(a(m+1)+m-1) \leq 3(2^{a-2} + 2^{a-3}) < 2^a + 2^{a-2}$.
+
+- If $b=m$, then $\text{Max}(a(m+1)+m) \leq 3 \cdot 2^{a-1} = 2^a + 2^{a-1}$.
+
+* Let $\langle n \rangle_m = 10^{m-1}10^{2m}w$, where the length of the word $w$ is $k - (3m+1)$. Since
+
+$$ (1 \ 1)M(r_s)M(r_{s-1}) = (1 \ 1)\binom{1 \ 0}{1 \ 1}\binom{2 \ 2}{1 \ 1} = 5(1 \ 1), $$
+
+we have $\text{Max}(k) = 5R^{(m)}(w) \leq 5\text{Max}(k - 3m - 1)$. We again express $k = a(m+1)+b$, where $b \in \{0, 1, \dots, m\}$. We have therefore
+
+$$ \text{Max}(a(m+1)+b) \leq 5\text{Max}((a-3)(m+1)+b+2). $$
+
+We distinguish:
+
+- If $b \in \{0, 1, \dots, m-2\}$, then using the induction hypothesis
+
+$$ \text{Max}(a(m+1)+b) \leq 5(2^{a-3} + 2^{a-4}) < 2^a, $$
+
+as required.
+
+- If $b=m-1$, then $\text{Max}(a(m+1)+m-1) \leq 5(2^{a-2}) = 2^a + 2^{a-2}$.
+
+- If $b=m$, then $\text{Max}(a(m+1)+m) \leq 5 \cdot 2^{a-2} < 2^a + 2^{a-1}$.
+
+* Let now $\langle n \rangle_m = 10^{m-1}10^{2m-1}w$. Claims 5.7, 5.8 and 5.9 imply that the word $w$ is of the form $w = 10^m \tilde{w}$ or $w = 10^{2m} \tilde{w}$. Thus we distinguish:
+
+- Let $\langle n \rangle_m = 10^{m-1}10^{2m-1}10^m \tilde{w}$. Since $(1\ 1)M(m-1)M(2m-1)M(m) = 8(1\ 1)$ and the length of the word $\tilde{w}$ is $k - 4m - 1$, we have
+
+$$ \text{Max}(k) \leq 8\text{Max}(k - 4m - 1) \leq 2^3\text{Max}(k - 3(m + 1)), $$
+
+which implies the desired result.
+
+- Let $\langle n \rangle_m = 10^{m-1}10^{2m-1}10^m \tilde{w}$. Since $(1\ 1)M(m-1)M(2m-1)M(2m) = 13(1\ 1)$ and the length of the word $\tilde{w}$ is $k - 5m - 1$, we have
+
+$$ \text{Max}(k) \leq 13\text{Max}(k - 5m - 1) \leq 2^4\text{Max}(k - 4(m + 1)), $$
+
+which implies the desired result.
+
+Note that Claim 5.8 is valid only for $m \geq 4$.
+
+$x=0$: Let us study the case $x=0$. We have to consider $y \in \{m, 2m-1, 2m, 3m-1\}$.

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+* Let $\langle n \rangle_m = 10^m w$. Since $(1\ 1)M(m) = 2(1\ 1)$, we have $\text{Max}(k) = 2\text{Max}(k - (m+1))$, what was to be proved.
+
+* Let $\langle n \rangle_m = 10^y w$, where $y \in \{2m-1, 2m, 3m-1\}$. In this case the complement of $n$ has the greedy expansion $\langle \bar{n} \rangle_m = 10^{m-1}\tilde{w}$ or $10^m\tilde{w}$, where $\tilde{w}$ is a greedy expansion of an integer. Such cases were already discussed before. Since $R^{(m)}(n) = R^{(m)}(\bar{n})$, the case is solved.
+
+This completes the proof for $m \ge 4$. Recall that the assumption $m \ge 4$ was used at one point of the discussion. For $m=3$ we have to consider $\langle n \rangle_m = 10^{m-1}10^{2m-1}10^{2m-1}\tilde{w} = 10^210^510^5\tilde{w}$. The discussion splits into cases according to the prefix of $\tilde{w}$. Necessarily, $\tilde{w} = (10^5)^k 10^{r_i} \cdots 10^{r_1}$ for some $k \ge 0$, $r_i \ne 5$. Since it has been shown that $r_1 \in \{3, 6\}$, we must have $i \ge 1$.
+
+If $r_i = 3$, then $\langle n \rangle_3$ contains the string $10^510^510^3$. The corresponding matrix is $M(5)M(5)M(3) = {8 \choose 5}$ which is majored by ${8 \choose 8} = M(3)M(3)M(3)$ corresponding to the string $10^310^310^310^3$ of the same length as $10^510^510^3$. Thus $10^510^510^3$ is forbidden for maximality.
+
+If $r_i = 6$, then $\langle n \rangle_3$ contains the string $10^510^510^6$. The corresponding matrix is $M(5)M(5)M(6) = {13 \choose 8}$ which is majored by ${16 \choose 8} = M(6)M(3)M(3)$ corresponding to the string $10^610^310^310^3$ of the same length as $10^510^510^6$. Thus $10^510^510^6$ is forbidden for maximality.
+
+If $r_i = 8$, then $\langle n \rangle_3$ contains the string $10^510^8$. The corresponding matrix is $M(5)M(8) = {7 \choose 4}$ which is majored by ${8 \choose 4} = M(6)M(3)M(3)$ corresponding to the string $10^610^310^3$ of the same length as $10^510^8$. Thus $10^510^8$ is forbidden for maximality.
+
+Last, if $r_i = 2$, then $\langle n \rangle_3$ contains the string $10^2(10^5)^j 10^2$ for some $j \ge 2$. The corresponding matrix is $M(2)M^2(5)M(2) = (F_{2j} F_{2j-1} \ F_{2j+1} F_{2j-2})$ which is majored by $(F_{2j+2} F_{2j+1} \ F_{2j+2} F_{2j-2}) = M^{j+1}(5)$ corresponding to the string $(10^5)^{j+1}$ of the same length as $10^2(10^5)^j 10^2$. Thus $10^2(10^5)^j 10^2$ is forbidden for maximality. $\square$
+
+# 6 Comments and open problems
+
+Although the numeration systems related to *m*-Bonacci numbers have been extensively studied from many different points of view, there remains a number of problems to be explored, even in the most simple Fibonacci case *m* = 2.
+
+One of these problems is to find a closed formula for the sequence *A(n*) giving the least integer having *n* representations as sums of distinct Fibonacci numbers, which is a sort of inverse to the function *R<sup>(2)</sup>(n)*. Some results about *A(n*) are given in [3, 4]. However, according to our knowledge, analogous function for *m*-Bonacci numeration system has never been studied.
+
+Another interesting question related to $R^{(2)}$ is the function rk($n$) defined in [5], counting the number of occurrences of a value $n$ among numbers $R^{(2)}(F_k)$, $R^{(2)}(F_k + 1)$, ..., $R^{(2)}(F_{k+1} - 1)$ for sufficiently large $k$. The authors of [5] show that the function is well defined, give a recurrent formula using the Euler function and exact value for $n$ prime. The function rk($n$) illustrates the exceptionality of the Fibonacci case $m=2$, because similar function cannot be defined if $m \ge 3$. We have already seen that the number of occurrences of the value $R^{(m)}(n) = 1$ among $R^{(m)}(F_k)$, $R^{(m)}(F_k + 1)$, ..., $R^{(m)}(F_{k+1} - 1)$ increases with $k$ to infinity. Similarly, it can be shown for other values $R^{(m)}(n)$.
+
+The literature often concentrates on the study of ambiguity in generalized Fibonacci numeration systems, where one allows coefficients only in $\{0, 1\}$. When omitting the limitation on

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+the coefficients, the problem becomes much more difficult. Even in case of the usual Fibonacci
+system, no compact formula is known for the so-called Fibagonci sequence $(B(n))_{n \in \mathbb{N}}$ counting
+the number of representations of $n$ as sum of (possibly repeating) Fibonacci numbers.
+
+One can also ask the question about numeration systems which allow coefficients ≥ 2 even in the greedy expansion of an integer. An example of these is the Ostrowski numeration system based on sequences defined by linear recurrences of second order with non-constant coefficients. Such numeration systems have been considered by Berstel [1] who shows that a formula similar to (5) is valid for counting the number of representations of $n$. Other properties of these numeration systems are to be explored.
+
+Acknowledgements
+
+The authors acknowledge partial support by Czech Science Foundation GA ČR 201/05/0169, and
+by the grant LC06002 of the Ministry of Education, Youth, and Sports of the Czech Republic.
+
+References
+
+[1] J. Berstel, *An excercise on Fibonacci representations*, RAIRO Theor. Inf. Appl. **35** (2001) 491–498.
+
+[2] M. Bicknell-Johnson, D. C. Fielder, *The number of representations of N using distinct Fibonacci numbers, counted by recursive formulas*, Fibonacci Quart. **37** (1999) 47–60.
+
+[3] M. Bicknell-Johnson, *The least integer having p Fibonacci representations, p prime*, Fibonacci Quart. **40** (2002) 260–265.
+
+[4] M. Bicknell-Johnson, *The smallest positive integer having F<sub>k</sub> representations as sums of distinct Fibonacci numbers*, in Applications of Fibonacci numbers **8** (1999) 47–52.
+
+[5] M. Edson, L. Zamboni, *On representations of positive integers in the Fibonacci base*, Theoret. Comput. Sci. **326** (2004), 241–260.
+
+[6] C. Frougny, *Fibonacci representations and finite automata*, IEEE Trans. Inform. Theory, **37** (1991), 393–399.
+
+[7] P.J. Grabner, P. Kirschenhofer, R.F. Tichy, *Combinatorial and arithmetical properties of linear numeration systems*, Combinatorica **22** (2002), 245–267.
+
+[8] P.J. Grabner, P. Liardet, R.F. Tichy, *Odometers and systems of numeration*, Acta Arith. **70** (1995), 103–123.
+
+[9] L.J. Guibas, A.M. Odlyzko, *String overlaps, pattern matching, and nontransitive games*, J. Combin. Theory Ser. A **30** (1981), 208–208.
+
+[10] W.H. Kautz, *Fibonacci codes for synchronization control*, IEEE Trans. Inform. Theory, **11** (1965), 284–292.
+
+[11] P. Kocábová, Z. Masáková, E. Pelantová, *Integers with a maximal number of Fibonacci representations*, RAIRO Theor. Inf. Appl. **39** (2005), 343–359.

samples/texts/3332461/page_2.md

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+The A 1g to B 1u and A 1g to B 2u transitions are symmetry forbidden and thus have a lower probability which is evident from the lowered intensity of their bands. The singlet A 1g to triplet B 1u transition is both symmetry forbidden and spin forbidden and therefore has the lowest intensity. This transition is forbidden by spin arguments ...
+
+## Raman scattering - Wikipedia
+
+Chapter 4 Symmetry and Group Theory 33 ... planes, C v. c. A screw has no symmetry operations other than the identity, for a C1 classification. d. The number 96 (with the correct type font) has a C2 axis perpendicular to the plane of the paper, making it C2h. e. Your choice-the list is too long to attempt to answer it here.
+
+## Selection rule - Wikipedia
+
+For the Love of Physics - Walter Lewin - May 16, 2011 - Duration: 1:01:26. Lectures by Walter Lewin. They will make you ? Physics. Recommended for you
+
+## Symmetry And Spectroscopy K V
+
+Chapter 7 - Symmetry and Spectroscopy - Molecular Vibrations - p. 1 -
+7. Symmetry and Spectroscopy - Molecular Vibrations 7.1 Bases for

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+molecular vibrations We investigate a molecule consisting of N atoms,
+which has 3N degrees of freedom. Taking ... Symmetry of wavefunction
+is equal to symmetry of Q k, i.e.
+
+**Symmetry: IR and Raman Spectroscopy**
+
+102 CHAPTER4. GROUPTHEORY In group theory, the elements considered are symmetry operations. For a given molecular system described by the Hamiltonian $H^$, there is a set of symmetry operations $O^i$ which commute with $H$: $O^i H^=O$.
+
+**Symmetry And Spectroscopy Of Molecules - K Veera Reddy ...**
+
+"The authors use an informal but highly effective writing style to present a uniform and consistent treatment of the subject matter." – Journal of Chemical Education. The primary focus of this text is to introduce students to vibrational and electronic spectroscopy, presenting applications of ...
+
+**Symmetry and Spectroscopy of Molecules: K. Veera Reddy ...**
+
+Raman scattering or the Raman effect / ? r ?? m ?n / is the inelastic scattering of photons by matter, meaning that there is an exchange of energy and a change in the light's direction. Typically this involves vibrational energy being gained by a molecule as incident photons from

samples/texts/3332461/page_4.md

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+a visible laser are shifted to lower energy.
+
+**K Veera Reddy - AbeBooks**
+
+where k is the bond force constant and m is the reduced mass for two nuclei of masses m1 and m2. $= 1 + 1$ m m1 m2 1 This yields the quantized vibrational level scheme shown in Figure 5.1 A. Because transitions between the v = 0 and v = 1 levels dominate in infrared or Raman spectroscopy, the harmonic
+
+**Group Theory in Spectroscopy - Elsevier**
+
+In vibrational spectroscopy, transitions are observed between different vibrational states. In a fundamental vibration, the molecule is excited from its ground state (v = 0) to the first excited state (v = 1). The symmetry of the ground-state wave function is the same as that of the molecule.
+
+**Infrared: Theory - Chemistry LibreTexts**
+
+How symmetric and asymmetric stretching of two identical groups can lead to two distinct signals in IR spectroscopy. Created by Jay. Watch the next lesson: h...
+
+**Electronic Spectroscopy: Interpretation - Chemistry LibreTexts**

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+R, the symmetry-defined types (irreducible representations, or irreps,
+in the language of group theory) ? i, and, lastly, the order of the
+group h (Table 7.1).
+
+Copyright code : [8722e4f451d30ae594c19503b1e9716d](https://www.citationstyle.org/8722e4f451d30ae594c19503b1e9716d)

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+HADWIGER'S THEOREM FOR DEFINABLE FUNCTIONS
+
+Y. BARYSHNIKOV, R. GHRIST, AND M. WRIGHT
+
+**ABSTRACT.** Hadwiger's Theorem states that $\mathbb{E}_n$-invariant convex-continuous valuations of definable sets in $\mathbb{R}^n$ are linear combinations of intrinsic volumes. We lift this result from sets to data distributions over sets, specifically, to definable $\mathbb{R}$-valued functions on $\mathbb{R}^n$. This generalizes intrinsic volumes to (dual pairs of) non-linear valuations on functions and provides a dual pair of Hadwiger classification theorems.
+
+# 1. INTRODUCTION
+
+Let $\mathbb{R}^n$ denote Euclidean $n$-dimensional space. A *valuation* on a collection $\mathcal{S}$ of subsets of $\mathbb{R}^n$ is an
+additive function $\nu : \mathcal{S} \to \mathbb{R}$:
+
+$$
+(1) \quad \nu(A) + \nu(B) = \nu(A \cap B) + \nu(A \cup B) \quad \text{whenever } A, B, A \cap B, A \cup B \in \mathcal{S}.
+$$
+
+Valuation v is $\mathbb{E}_n$-invariant if $v(\varphi A) = v(A)$ for all $A \in S$ and $\varphi \in \mathbb{E}_n$, the group of Euclidean (or rigid) motions in $\mathbb{R}^n$. A classical theorem of Hadwiger [16] states that the $\mathbb{E}_n$-invariant continuous valuations on compact convex sets $S$ in $\mathbb{R}^n$ (here a valuation is *continuous* with respect to convergence of sets in the Hausdorff metric) form a finite-dimensional $\mathbb{R}$-vector space generated by intrinsic volumes $\mu_k$, $k = 0, \dots, n$.
+
+**Theorem 1** (Hadwiger). Any $\mathbb{E}_n$-invariant continuous valuation $v$ on compact convex subsets of $\mathbb{R}^n$ is a linear combination of the intrinsic volumes:
+
+$$
+(2) \qquad \nu = \sum_{k=0}^{n} c_k \mu_k,
+$$
+
+for some constants $c_k \in \mathbb{R}$. If $\nu$ is homogeneous of degree $k$, then $\nu = c_k \mu_k$.
+
+The intrinsic volumes¹ $\mu_k$ are characterized uniquely by (1) $\mathbb{E}_n$ invariance, (2) normalization with respect to a closed unit ball, and (3) homogeneity: $\mu_k(\lambda \cdot A) = \lambda^k(A)$ for all $A \in \mathcal{S}$ and $\lambda \in \mathbb{R}^+$. These measures generalize Euclidean $n$-dimensional volume ($\mu_n$) and Euler characteristic ($\mu_0$).
+
+This paper extends Hadwiger's Theorem to similar valuations on functions instead of sets. Section 2 gives background on the definable (o-minimal) setting that lifts Hadwiger's Theorem to tame, non-convex sets and then to constructible functions; there, we also review the convex-geometric, integral-geometric, and sheaf-theoretic approaches to Hadwiger's Theorem. In Section 4, we consider definable functions Def($\mathbb{R}^n$) as $\mathbb{R}$-valued functions with tame graphs, and correspondingly define dual pairs of (typically) non-linear "integral" operators $\int \cdot |\mathrm{d}\mu_k|$ and $\int \cdot [\mathrm{d}\mu_k]$ mapping Def($\mathbb{R}^n$) $\to \mathbb{R}$ as generalizations of intrinsic volumes, so that $\int 1_A [\mathrm{d}\mu_k] = \mu_k(A) = \int 1_A [\mathrm{d}\mu_k]$ for
+
+Key words and phrases. valuations, Hadwiger measure, intrinsic volumes, Euler characteristic.
+This work supported by DARPA # HR0011-07-1-0002 and by ONR N000140810668.
+
+¹Intrinsic volumes are also known in the literature as *Hadwiger measures*, *quermassintegrale*, *Lipschitz-Killing curvatures*, *Minkowski functionals*, and, likely, more.

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+FIGURE 1. Conormal cycles of the point $p$, the open interval $(a, b)$, and the closed interval $[c, d]$ illustrate the additivity of the conormal cycle.
+
+**Continuity:** The flat norm on conormal cycles yields a topology on definable subsets on which the intrinsic volumes are continuous. For definable subsets A and B, define the *flat metric* by
+
+$$ (13) \qquad d_b(A, B) = |(\mathbf{C}^A - \mathbf{C}^B) \cap B_1^*\mathbb{R}^n|_b, $$
+
+thereby inducing the *flat topology*. (That this is a metric follows from $\mathbf{C}^-$ being an injection on definable subsets.) For any $T \in \Omega_n$ and $\omega \in \Omega_c^n$, both supported on $B_1^*\mathbb{R}^n$:
+
+$$ (14) \qquad |T(\omega)| \le |T|_b \cdot \max \left\{ \sup_{B_1^*\mathbb{R}^n} |\omega|, \sup_{B_1^*\mathbb{R}^n} |d\omega| \right\}. $$
+
+Since the intrinsic volumes can be represented by integration of bounded forms over the intersection of the conormal cycle with the unit ball bundle, the intrinsic volumes are continuous with respect to the flat topology. We remark also that for the *convex* constructible sets, the flat topology is equivalent to the one given by the Hausdorff metric.
+
+### 3. INTRINSIC VOLUMES FOR CONSTRUCTIBLE FUNCTIONS
+
+It is possible to extend the intrinsic volumes beyond definable sets. The *constructible functions*, CF, are functions $h: \mathbb{R}^n \to \mathbb{R}$ with discrete image and definable level sets. By abuse of terminology, CF will always refer to compactly supported definable functions with *finite* image in $\mathbb{R}$.
+
+As the integral with respect to the Euler characteristics is well defined for constructible functions, one can extend the intrinsic volumes to constructible functions using the slicing definition above:
+
+$$ (15) \qquad \mu_k(h) = \int_{S_{n,n-k}} \int_{\mathbb{R}^n/L} \left( \int_{L+x} h d\chi \right) dx d\gamma(L). $$
+
+In so doing, one obtains, e.g., the following generalization of the Poincaré theorem for Euler characteristic.

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+We need the Verdier duality operator in CF, which is defined e.g. in [25]. Briefly, the dual of $h \in \mathbf{CF}$ is a function $\mathbf{D}h$ whose value at $x_0$ is given by
+
+$$ (16) \qquad (\mathbf{D}h)(x_0) = \lim_{\epsilon \to 0} \int_{\mathbb{R}^n} \mathbf{1}_{B(x_0, \epsilon)} h \, d\chi, $$
+
+where the integral is with respect to Euler characteristic (see also [6]), and $B(x_0, \epsilon)$ is the n-dimensional ball of radius $\epsilon$ centered at $x_0$. In many cases, this duality swaps interiors and closures. For example, if A is a convex open set with closure $\bar{A}$, then $\mathbf{D1}_A = \mathbf{1}_{\bar{A}}$ and $\mathbf{D1}_{\bar{A}} = \mathbf{1}_A$.
+
+**Proposition 4.** For a constructible function $h$ on $\mathbb{R}^n$, $h \in \mathbf{CF}$, and $\mathbf{D}$ the Verdier duality operator in CF,
+
+$$ (17) \qquad \int_{\mathbb{R}^n} h \, d\mu_k = (-1)^{n-k} \int_{\mathbb{R}^n} \mathbf{D}h \, d\mu_k. $$
+
+*Proof.* The result holds in the case $k=0$ (see [25]). From Equation (15), $\mu_k$ is defined by integration with respect to $d\chi$ along codimension-k planes, followed by the integration over the planes. By Sard's theorem, for (Lebesgue) almost all $L \in \mathcal{S}_{n,n-k}$ and $x \in \mathbb{R}^n/L$, the level sets of $h$ are transversal to $L+x$, whence, by Thom's second isotopy lemma, [28],
+
+$$ (18) \qquad \int_{L+x} h \, d\chi = (-1)^{n-k} \int_{L+x} \mathbf{D}h \, d\chi $$
+
+for almost all L and x. Integration over $\mathcal{P}_{n,n-k}$ finishes the proof. $\square$
+
+*Remark 5.* If definable sets $A, \bar{A} \subset \mathbb{R}^n$ satisfy $\mathbf{D1}_A = \mathbf{1}_{\bar{A}}$, then Proposition 4 implies
+
+$$ (19) \qquad \mu_k(A) = (-1)^{n-k} \mu_k(\bar{A}). $$
+
+### 4. INTRINSIC VOLUMES FOR DEFINABLE FUNCTIONS
+
+The next logical step, lifting from constructible to definable functions, is the focus of this paper. Let $\textbf{Def}(\mathbb{R}^n)$ denote the *definable functions* on $\mathbb{R}^n$, that is, the set of functions $h : \mathbb{R}^n \to \mathbb{R}$ whose graphs are definable sets in $\mathbb{R}^n \times \mathbb{R}$ which coincide with $\mathbb{R}^n \times \{0\}$ outside of a ball (thus compactly supported and bounded). In [7], integration with respect to Euler characteristic $\mu_0$ was lifted to a dual pair of nonlinear “integrals” $\int \cdot |\mathrm{d}\chi|$ and $\int \cdot [\mathrm{d}\chi]$ via the following limiting process, now extended to $\mu_k$:
+
+**Definition 6.** For $h \in \textbf{Def}(\mathbb{R}^n)$, the *lower* and *upper* Hadwiger integrals of $h$ are, respectively,
+
+$$ (20) \qquad \begin{aligned} \int h \rfloor d\mu_k &= \lim_{m \to \infty} \frac{1}{m} \int |\mathrm{mh}| \, d\mu_k, \text{ and} \\ \int h \lceil d\mu_k \rceil &= \lim_{m \to \infty} \frac{1}{m} \int [\mathrm{mh}] \, d\mu_k. \end{aligned} $$
+
+For $k=n$ these two definitions agree with each other and with the Lebesgue integral; for all $k<n$, they differ. For $k=0$, these become the definable Euler integrals $\int \cdot |\mathrm{d}\chi|$ and $\int \cdot [\mathrm{d}\chi]$. The following result demonstrates several equivalent formulations, mirroring those of Section 2. As a consequence, the limits in Definition 6 are well-defined, following from compact support and the well-definedness of $|\mathrm{d}\chi|$ and $\lceil\mathrm{d}\chi\rceil$ from [7].

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+FIGURE 2. The lower Hadwiger integral is defined as a limit of lower step functions (left), as in Definition 6. It can also be expressed in terms of excursion sets (right), as in Theorem 7, equation (21).
+
+5. CONTINUOUS VALUATIONS
+
+Valuations on functions are a straightforward generalization of valuations on sets. A *valuation* on $\mathbf{Def}(\mathbb{R}^n)$ is a functional $v : \mathbf{Def}(\mathbb{R}^n) \to \mathbb{R}$, satisfying $v(0) = 0$ and the following additivity condition:
+
+$$ (27) \qquad v(f) + v(g) = v(f \vee g) + v(f \wedge g), $$
+
+where $\vee$ and $\wedge$ denote the pointwise max and min, respectively.
+
+We present two useful topologies on $\mathbf{Def}(\mathbb{R}^n)$ that allow us to consider continuous valuations. With these topologies, the notion of a continuous valuation on $\mathbf{Def}(\mathbb{R}^n)$ properly extends the notion of a continuous valuation on definable subsets of $\mathbb{R}^n$.
+
+**Definition 8.** Let $f, g \in \mathbf{Def}(\mathbb{R}^n)$. The lower and upper flat metrics on definable functions, denoted $\underline{d}_b$ and $\overline{d}_b$, respectively, are defined as follows (see 13):
+
+$$ (28) \qquad \underline{d}_b(f, g) = \int_{-\infty}^{\infty} d_b(C^{[f \ge s]}, C^{[g \ge s]}) \, ds \quad \text{and} $$
+
+$$ (29) \qquad \overline{d}_b(f, g) = \int_{-\infty}^{\infty} d_b(C^{[f>s]}_-, C^{[g>s]}_-) \, ds. $$
+
+The distinct topologies induced by the lower and upper flat metrics are the *lower* and *upper flat topologies* on definable functions. A valuation on definable functions is *lower-* or *upper-continuous* if it is continuous in the lower or upper flat topology, respectively.
+
+Note that the integrals in (28) and (29) are well-defined because they may be written with finite bounds, as it suffices to integrate between the minimum and maximum values of f and g. These metrics extend the flat metric on definable sets, for they reduce to (13) when f and g are characteristic functions.
+
+*Remark 9.* Definition 8 does result in *metrics*. If $\underline{d}_b(f,g) = 0$, then $|C^{[f \ge s]} - C^{[g \ge s]}|_b = 0$ only for $s$ in a set of Lebesgue measure zero. However, if the excursion sets of f and g agree almost everywhere, then all excursion sets of f and g agree, and thus $f = g$. For, if $\{s_i\}_i$ is a sequence of negative real numbers converging 0, and $\{f \ge s_i\} = \{g \ge s_i\}$ for all i, then:
+
+$$ \{f \ge 0\} = \bigcap_i \{f \ge s_i\} = \bigcap_i \{g \ge s_i\} = \{g \ge 0\}. $$

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+The result for $\bar{d}_b(f, g)$ follows similarly from the observation that $\{f > 0\} = \bigcup_{s>0}\{f > s\}$.
+
+*Remark 10.* That the lower and upper flat topologies are distinct can be seen by noting that for the identity function $f$ on the interval $[0, 1]$, the sequence of lower step functions $g_m = \frac{1}{m!} \lfloor mf \rfloor$ converges (as $m \to \infty$) to $f$ in the lower flat topology, but not in the upper flat topology. Dually, upper step functions converge in the upper flat topology, but not in the lower.
+
+**Lemma 11.** The lower and upper Hadwiger integrals are lower- and upper-continuous, respectively.
+
+*Proof.* Let $f, g \in \text{Def}(\mathbb{R}^n)$ be supported on $X \subset \mathbb{R}^n$. The following inequality for the lower integrals is via (14):
+
+$$ 
+\begin{aligned}
+\left|\int f \rfloor d\mu_k\rfloor - \int g \rfloor d\mu_k\rfloor\right| &= \left|\int_{-\infty}^{\infty} (\mu_k\{f \ge s\} - \mu_k\{g \ge s\}) ds\right| \\
+&\le \int_{-\infty}^{\infty} \left|\left(C^{f \ge s}\right) - \left(C^{g \ge s}\right)\right| \cap B_1^*\mathbb{R}^n|_b \cdot \max \left\{\sup_{B_1^*\mathbb{R}^n} |\omega|, \sup_{B_1^*\mathbb{R}^n} |d\omega|\right\} \\
+&= \underline{d}_b(f, g) \cdot \max \left\{\sup_{B_1^*\mathbb{R}^n} |\omega|, \sup_{B_1^*\mathbb{R}^n} |d\omega|\right\}
+\end{aligned}
+$$
+
+Since $\omega_k$ and $d\omega_k$ are bounded, we have continuity of the lower integrals in the lower flat topology. The proof for the upper integrals is analogous. $\square$
+
+For *constructible* functions, the lower and upper flat topologies of the previous section are equivalent. Thus, we may refer to the *flat topology* on constructible functions without specifying *upper* or *lower*. A valuation on constructible functions is *conormal continuous* if it is continuous with respect to the flat topology. Conormal continuity is the same as “smooth” in the Alesker sense [3, 4], but distinct from continuity in the topology induced by the Hausdorff metric on definable sets.
+
+## 6. HADWIGER'S THEOREM FOR FUNCTIONS
+
+A dual pair of Hadwiger-type classifications for (lower-/upper-) continuous Euclidean-invariant valuations is the goal of this paper.
+
+**Lemma 12.** If $\nu: \text{CF}(\mathbb{R}^n) \to \mathbb{R}$ is a (conormal) continuous valuation on constructible functions, invariant with respect to the right action by Euclidean motions, then $\nu$ is of the form:
+
+$$ \nu(h) = \sum_{k=0}^{n} \int_{\mathbb{R}^n} c_k(h) d\mu_k. $$
+
+for some coefficient functions $c_k : \mathbb{R} \to \mathbb{R}$ with $c_k(0) = 0$.
+
+*Proof.* For the class of indicator functions for convex sets $\{h = r \cdot 1_A : r \in \mathbb{Z}\}$ and $A \subset \mathbb{R}^n$ definable, continuity of $\nu$ in the flat topology implies that $\nu$ is continuous in the Hausdorff topology. Since convex tame sets are dense (in Hausdorff metric) among convex sets in $\mathbb{R}^n$, Hadwiger's Theorem for sets implies that
+
+$$ (30) \qquad \nu(r \cdot 1_A) = \sum_{k=0}^{n} c_k(r) \mu_k(A), $$

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+where $c_k(r)$ are constants that depend only on $v$, not on $A$. Conormal continuity implies that the valuation $v(A)$ is the integral of the linear combination of the forms $\omega_k$ (defined in (12)),
+
+$$ (31) \qquad \sum_{k=0}^{n} c_k(r) \alpha_k $$
+
+over $\mathbb{C}^n$.
+
+Now suppose $h = \sum_{i=1}^{m} r_i 1_{A_i}$ is a finite sum of indicator functions of disjoint definable subsets $A_1, \dots, A_m$ of $\mathbb{R}^n$ for some integer constants $r_1 < r_2 < \dots < r_m$. By equation (30) and additivity,
+
+$$ (32) \qquad v(h) = \sum_{k=0}^{n} \sum_{i=1}^{m} c_k(r_i) \mu_k(A_i). $$
+
+We can rewrite equation (32) in terms of excursion sets of $h$. Let $B_i = \cup_{j \ge i} A_j$. That is, $B_i = \{h \ge r_i\}$ and $B_i = \{h > r_{i-1}\}$. Then the valuation $v(h)$ can be expressed as:
+
+$$ (33) \qquad v(h) = \sum_{k=0}^{n} \sum_{i=1}^{m} (c_k(r_i) - c_k(r_{i-1})) \mu_k(B_i), $$
+
+where $c_k(r_0) = 0$. Thus, a valuation of a constructible function can be expressed as a sum of finite differences of valuations of its excursion sets. Equivalently, equation (33) can be written in terms of constructible Hadwiger integrals:
+
+$$ (34) \qquad v(h) = \sum_{k=0}^{n} \int_{\mathbb{R}^n} c_k(h) \, d\mu_k. $$
+
+Since we require that a valuation of the zero function is zero, it must be that $c_k(0) = 0$ for all $k$. $\square$
+
+Note that Lemma 12 holds for functions of the form $h = \sum_{i=1}^{m} r_i 1_{A_i}$ where the $A_i$ are definable and the $r_i \in \mathbb{R}$ are not necessarily integers.
+
+In writing an arbitrary valuation on definable functions as a sum of Hadwiger integrals, the situation becomes complicated if the coefficient functions $c_k$ are decreasing on any interval. The following proposition illustrates the difficulty:
+
+**Proposition 13.** Let $c: \mathbb{R} \to \mathbb{R}$ be a continuous, strictly decreasing function. Then,
+
+$$ (35) \qquad \lim_{m \to \infty} \int_{\mathbb{R}^n} c\left(\frac{1}{m} \lceil mh \rceil\right) \, d\mu_k = \lim_{m \to \infty} \int_{\mathbb{R}^n} \frac{1}{m} \lfloor mc(h) \rfloor \, d\mu_k. $$
+
+*Proof.* On the left side of equation (35), we integrate $c$ composed with upper step functions of $h$:
+
+$$ \int_{\mathbb{R}^n} c\left(\frac{1}{m}\lceil mh \rceil\right) d\mu_k = \sum_{i \in \mathbb{Z}} c\left(\frac{i}{m}\right) \cdot \mu_k\left\{\frac{i-1}{m} < h \le \frac{i}{m}\right\}. $$
+
+On the right side of equation (35), we integrate lower step functions of the composition $c(h)$:
+
+$$ \int_{\mathbb{R}^n} \frac{1}{m} [\mathrm{mc}(h)] d\mu_k = \sum_{t \in \mathbb{Z}} \frac{t}{m} \cdot \mu_k\left\{\frac{t}{m} \le c(h) < \frac{t+1}{m}\right\}. $$
+
+Since $c$ is strictly decreasing, $c^{-1}$ exists. There exists a discrete set
+
+$$ S = \left\{ c^{-1}\left(\frac{t}{m}\right) \mid t \in \mathbb{Z} \right\} \cap \{\text{neighborhood around range of } h\}. $$

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+We can alternately express equation (37) as
+
+$$
+(38) \qquad v(h_m) = \sum_{k=0}^{n} \int_{\mathbb{R}^n} c_k(h_m) \lfloor d\mu_k \rfloor,
+$$
+
+where we choose lower rather than upper integrals since $v$ is continuous in the lower flat topology.
+Continuity of $v$, and convergence of $h_m$ to $h$, in the lower flat topology imply that $v(h_m)$ converges
+to $v(h)$ as $h \to \infty$. More specifically,
+
+$$
+(39) \qquad v(h) = \lim_{m \to \infty} v(h_m) = \sum_{k=0}^{n} \lim_{m \to \infty} \int_{\mathbb{R}^n} c_k(h_m) \lfloor d\mu_k \rfloor.
+$$
+
+By continuity of the lower Hadwiger integrals (Lemma 11) and the $c_k$, Equation (39) becomes
+
+$$
+(40) \qquad v(h) = \sum_{k=0}^{n} \int_{\mathbb{R}^n} c_k \left( \lim_{m \to \infty} h_m \right) \lfloor d\mu_k \rfloor = \sum_{k=0}^{n} \int_{\mathbb{R}^n} c_k(h) \lfloor d\mu_k \rfloor.
+$$
+
+The proof for the upper valuation is analogous.
+
+**Corollary 15.** Any $\mathbb{E}_n$-invariant valuation both upper- and lower-continuous is a weighted Lebesgue integral.
+
+*Proof.* Integration with respect to $\lfloor d\mu_k \rfloor$ and $\lceil d\mu_j \rceil$ are independent unless $k = j = n$. For any $v$
+both upper- and lower-continuous, we have
+
+$$
+v(h) = \sum_{k=0}^{n} \int_{\mathbb{R}^n} c_k(h) \lfloor d\mu_k \rfloor = \sum_{k=0}^{n} \int_{\mathbb{R}^n} \bar{c}_k(h) \lceil d\mu_k \rceil
+$$
+
+for some functions $c_k$ and $\bar{c}_k$.
+
+Lower and upper Hadwiger integrals with respect to $\mu_k$ are unequal, except when $k=n$, implying
+that $\underline{c}_k = \bar{c}_k = 0$ for $k=0, 1, \dots, n-1$, and $\overline{c}_n = \bar{c}_n$. Therefore,
+
+$$
+v(h) = \int_{\mathbb{R}^n} c(h) \, d\mu_n
+$$
+
+for some continuous function $c : \mathbb{R} \rightarrow \mathbb{R}$, and with $d\mu_n = [\mathrm{d}\mu_n] = [\mathrm{d}\mu_n]$ denoting Lebesgue measure. $\square$
+
+7. SPECULATION
+
+The present constructions are potentially applicable to generalizations of current applications of
+intrinsic volumes. One such recent application is to the dynamics of cellular structures, such as
+crystals and foams in microstructure of materials. The cells in such structures often change shape
+and size over time in order to minimize the total energy level in the system. Let $C = \bigcup_{i=0}^{n} C_i$ be a
+closed $n$-dimensional cell, with $C_i$ denoting the union of all $i$-dimensional features of the cell: *i.e.*,  
+$C_0$ is the set of vertices, $C_1$ the set of edges, etc. MacPherson and Srolovitz found that when the  
+cell structure changes by a process of *mean curvature flow*, the volume of the cell changes according  
+to
+
+$$
+(41) \qquad \frac{\mathrm{d}\mu_n}{\mathrm{d}t}(C) = -2\pi M\gamma \left( \mu_{n-2}(C_n) - \frac{1}{6}\mu_{n-2}(C_{n-2}) \right)
+$$
+
+where M and γ are constants determined by the material properties of the cell structure [22].
+Replacing the intrinsic volumes of cells with Hadwiger integrals may (1) lead to interesting dynamical systems on the (singular) foliations (by the level sets of a piece-wise smooth function, and

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+(2) allow for description of evolution of real-valued physical fields (temperature, density, etc.) of cells.
+
+A more widely-known application of the intrinsic volumes is in the formulas for expected Euler characteristic of excursion sets in Gaussian random fields [1, 2]. These formulae and the associated Gaussian kinematic formula [2] rely crucially on the intrinsic volumes of excursion sets. It is already recognized in recent work [8] that the definable Euler measure $[d\chi] = [d\mu_0]$ is relevant to Gaussian random fields: we strongly suspect that the other definable Hadwiger measures $[d\mu_k]$ and $[d\mu_k]$ of this paper are immediately applicable to Gaussian random fields.
+
+## REFERENCES
+
+[1] R. Adler, *The Geometry of Random Fields*, Wiley, 1981; reprinted by SIAM, 2009.
+
+[2] R. Adler and J. Taylor, "Topological Complexity of Random Functions", Springer Lecture Notes in Mathematics, Vol. 2019, Springer, 2011.
+
+[3] S. Alesker, "Theory of valuations on manifolds: a survey," *Geometric and Functional Analysis*, **17**(4), 2007, 1321–1341.
+
+[4] S. Alesker, "Valuations on manifolds and integral geometry," *Geometric and Functional Analysis*, **20**(5), 2010, 1073–1143.
+
+[5] A. Bernig, "Algebraic Integral Geometry," *Global Differential Geometry*, edited by C Bär, J. Lohkamp, and M. Schwarz, Springer, 2012.
+
+[6] Y. Baryshnikov and R. Ghrist, "Target enumeration via Euler characteristic integration," *SIAM J. Appl. Math.*, **70**(3), 2009, 825–844.
+
+[7] Y. Baryshnikov and R. Ghrist, "Definable Euler integration," *Proc. Nat. Acad. Sci.*, **107**(21), May 25, 9525-9530, 2010.
+
+[8] O. Bobrowski and M. Strom Borman, "Euler Integration of Gaussian Random Fields and Persistent Homology," 2011, arXiv:1003.5175.
+
+[9] J. Cheeger, W. Müller, and R. Schrader, "On the curvature of piecewise flat spaces," *Comm. Math. Phys.* **92**(3), 1984, 405–454.
+
+[10] M. Coste, An Introduction to o-minimal Geometry, Dip. Mat. Univ. Pisa, Dottorato di Ricerca in Matematica, Istituti Editoriali e Poligrafici Internazionali, Pisa, 2000, http://www.ihp-raag.org/publications.php.
+
+[11] H. Federer, *Geometric Measure Theory*, Springer 1969.
+
+[12] J. Fu, "Curvature measures of subanalytic sets", Amer. J. Math., **116**, (1994), 819-890.
+
+[13] J. Fu, "Notes on Integral Geometry," 2011, http://www.math.uga.edu/~fu/notes.pdf.
+
+[14] R. Ghrist and M. Robinson, "Euler-Bessel and Euler-Fourier transforms," *Inv. Prob.*, to appear.
+
+[15] Guesin-Zade, "Integration with respect to the Euler characteristic and its applications," Russ. Math. Surv., **65**:3, 2010, 399–432.
+
+[16] H. Hadwiger, "Integralsätze im Konvexring," Abh. Math. Sem. Hamburg, **20**, 1956, 136–154.
+
+[17] D. A. Klain and G.-C. Rota, *Introduction to Geometric Probability*, Cambridge, 1997.
+
+[18] M. Kashiwara, "Index theorem for constructible sheaves," *Astrisque*, **130**, 1985, 193–209.
+
+[19] D. A. Klain, "A Short Proof of Hadwiger's Characterization Theorem," Mathematika, **42**, 1995, 329–339.
+
+[20] M. Kashiwara and P. Schapira, *Sheaves on Manifolds*, Springer, 1990.
+
+[21] M. Ludwig, "Valuations on function spaces," Adv. Geom., **11**, (2011), 745–756.
+
+[22] R. D. MacPherson and D. J. Srolovitz, "The von Neumann relation generalized to coarsening of three-dimensional microstructures," Nature, **446**, 2007, 1053–1055.
+
+[23] L. I. Nicolaescu, "Conormal Cycles of Tame Sets," preprint, 2010, http://www.nd.edu/~lnicolae/conormal.pdf.
+
+[24] L. I. Nicolaescu, "On the Normal Cycles of Subanalytic Sets," Ann. Glob. Anal. Geom., **39**, 2011, 427–454.
+
+[25] P. Schapira, "Operations on constructible functions," J. Pure Appl. Algebra, **72**, 1991, 83–93.
+
+[26] J. Schürmann, *Topology of Singular Spaces and Constructible Sheaves*, Birkhäuser, 2003.
+
+[27] S. H. Schanuel, "What is the Length of a Potato?" in Lecture Notes in Mathematics, Springer, 1986, 118–126.
+
+[28] M. Shiota, *Geometry of subanalytic and semialgebraic sets*, Birkhäuser, 1997.
+
+[29] L. Van den Dries, *Tame Topology and O-Minimal Structures*, Cambridge University Press, 1998.

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+DEPARTMENTS OF MATHEMATICS AND ELECTRICAL AND COMPUTING ENGINEERING, UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN, URBANA IL, USA
+
+*E-mail address:* ymb@uiuc.edu
+
+DEPARTMENTS OF MATHEMATICS AND ELECTRICAL/SYSTEMS ENGINEERING, UNIVERSITY OF PENNSYLVANIA, PHILADELPHIA PA, USA
+
+*E-mail address:* ghrist@math.upenn.edu
+
+DEPARTMENT OF MATHEMATICS, HUNTINGTON UNIVERSITY, HUNTINGTON IN, USA
+
+*E-mail address:* mwright@huntington.edu

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+all A definable. These integrals are $\mathbb{E}_n$-invariant and satisfy generalized homogeneity and additivity conditions reminiscent of intrinsic volumes; they are furthermore compact-continuous with respect to a dual pair of topologies on functions. This culminates in Section 6 in a generalization of Hadwiger's Theorem for functions:
+
+**Theorem 2 (Main).** Any $\mathbb{E}_n$-invariant definably lower- (resp. upper-) continuous valuation $v : \text{Def}(\mathbb{R}^n) \to \mathbb{R}$ is of the form:
+
+$$ (3) \qquad v(h) = \sum_{k=0}^{n} \left( \int_{\mathbb{R}^n} c_k \circ h \rfloor d\mu_k \right) $$
+
+ resp., integrals with respect to $d\mu_k$) for some $c_k \in C(\mathbb{R})$ continuous and monotone, satisfying $c_k(0) = 0$.
+
+The $k=0$ intrinsic measure $d\mu_0$ is a recent generalization of $d\chi$ of Euler characteristic [7] shown to have applications to signal processing [14] and Gaussian random fields [8]. The $k=n$ intrinsic measure $d\mu_n$ is Lebesgue volume. The measures come in dual pairs $d\mu_k$ and $d\mu_k$ as a manifestation of (Verdier-Poincaré) duality. Our results yield the following:
+
+**Corollary 3.** Any $\mathbb{E}_n$-invariant valuation, both upper- and lower-continuous, is a weighted Lebesgue integral.
+
+We conclude the Introduction remarking that over the past few years several very interesting papers by M. Ludwig, A. Tsang and others appeared, that deal with valuations (often tensor- or set-valued) on various functional spaces (such as $L_p$ and Sobolev spaces): for a recent report, see [21]. Our approach deviates from this circle of results primarily in the choice of the functional space: definable functions form a distinctly different domain for the valuation. The quite fine topologies we impose on the definable functions yield a rich supply of the valuations continuous in these topologies.
+
+## 2. BACKGROUND
+
+2.1. **Euler characteristic.** Intrinsic volumes are built upon the Euler characteristic. Among the many possible approaches to this topological invariant — combinatorial [17], cohomological [15], sheaf-theoretic [25, 26], we use the language of o-minimal geometry [29]. An *o-minimal structure* is a sequence $\O = (O_n)_n$ of Boolean algebras of subsets of $\mathbb{R}^n$ which satisfy a few basic axioms (closure under cross products and projections; algebraic basis; and $O_1$ consists of finite unions of points and open intervals). Examples of o-minimal structures include the semialgebraic sets, globally subanalytic sets, and (by slight abuse of terminology) semilinear sets; more exotic structures with exponentials also occur [29]. The details of o-minimal geometry can be ignored in this paper, with the following exceptions:
+
+(1) Elements of $\O$ are called *tame* or, more properly, *definable* sets.
+
+(2) A mapping between definable sets is definable if and only if its graph is a definable set.
+
+(3) The basic equivalence relation on definable sets is definable bijection; these are not necessarily continuous.
+
+(4) The Triangulation Theorem [29, Thm 8.1.7, p. 122]: any definable set $Y$ is definably equivalent to a finite disjoint union of open simplices $\{\sigma\}$ of different dimensions.
+
+(5) The Hardt Theorem [29]: for $f: X \to Y$ definable, $Y$ has a triangulation into open simplices $\{\sigma\}$ such that $f^{-1}(\sigma)$ is homeomorphic to $U_\sigma \times \sigma$ for $U_\sigma$ definable, and, on this inverse image, $f$ acts as projection.

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+For more information, the reader is encouraged to consult [29, 10, 24].
+
+The (o-minimal) Euler characteristic is the valuation $\chi$ that evaluates to $(-1)^k$ on an open k-simplex.
+It is well-defined and invariant under definable bijection [29, Sec. 4.2], and, among definable sets
+of fixed dimension (the dimension of the largest cell in a triangulation), is a complete invariant
+of definable sets up to definable bijection. Note that the o-minimal Euler characteristic coincides
+with the homological Euler characteristic (alternating sum of ranks of homology groups) on com-
+pact definable sets. For locally compact definable sets, it has a cohomological definition (alternat-
+ing sum of ranks of compactly-supported sheaf cohomology), yielding invariance with respect to
+proper homotopy.
+
+**2.2. Intrinsic volumes.** Intrinsic volumes have a rich history (see, e.g., [5, 9, 17, 27]) and as many formulations as names, including the following:
+
+**Slices:** One way to define the intrinsic volume $\mu_k(A)$ of a definable set $A$ is in terms of the Euler characteristic of all slices of $A$ along affine codimension-k planes:
+
+$$
+(4) \qquad \mu_k(A) = \int_{\mathcal{P}_{n,n-k}} \chi(A \cap P) \, d\lambda(P),
+$$
+
+where $\lambda$ is the following measure on $\mathcal{P}_{n,n-k}$, the space of affine $(n-k)$-planes in $\mathbb{R}^n$. Each affine subspace $P \in \mathcal{P}_{n,n-k}$ is a translation of some linear subspace $L \in \mathcal{G}_{n,n-k}$, the Grassmannian of $(n-k)$-dimensional subspaces of $\mathbb{R}^n$. That is, $P$ is uniquely determined by $L$ and a vector $\mathbf{x} \in L^\perp$, such that $P = L + \mathbf{x}$. Thus, we can integrate over $\mathcal{P}_{n,n-k}$ by first integrating over $L^\perp$ and then over $\mathcal{G}_{n,n-k}$. Equation (4) is equivalent to
+
+$$
+(5) \qquad \mu_k(A) = \int_{\mathcal{G}_{n,n-k}} \left( \int_{\mathbb{R}^n/L} \chi(A \cap (L+x)) \, dx \right) d\gamma(L),
+$$
+
+where $L \in \mathcal{G}_{n,n-k}$, the factorspace $\mathbb{R}^n/L$ is given the natural Lebesgue measure, and $\gamma$ is the Haar (i.e. SO(n)-invariant) measure on the Grassmannian, scaled appropriately.
+
+**Projections:** Dual to the above definition, one can express $\mu_k$ in terms of projections onto k-dimensional linear subspaces: for any definable $A \subset \mathbb{R}^n$ and $0 \le k \le n$,
+
+$$
+(6) \qquad \mu_k(A) = \int_{\mathcal{G}_{n,k}} \left( \int_L \chi(\pi_L^{-1}(x)) \right) dx d\gamma(L)
+$$
+
+where $L \in \mathcal{G}_{n,k}$ and $\pi_L^{-1}(x)$ is the fiber over $x \in L$ of the orthogonal projection map $\pi : A \to L$. For
+A convex, Equation (6) reduces to
+
+$$
+\mu_k(A) = \int_{\mathcal{G}_{n,k}} \mu_k(A|L) \, d\gamma(L)
+$$
+
+where the integrand is the k-dimensional (Lebesgue) volume of the projection of A onto a k-dimensional subspace L of $\mathbb{R}^n$.
+
+**2.3. Normal, conormal, and characteristic cycles.** Perspectives from geometric measure theory and sheaf theory are also relevant to the definition of intrinsic volumes. In this section, we restrict to the o-minimal structure of globally subanalytic sets and use analytic tools based on geometric measure theory, following Alesker [3, 4], Fu [12], Nicolaescu [23, 24] and many others.
+
+Let $\Omega_c^k(\mathbb{R}^n)$ be the space of differential k-forms on $\mathbb{R}^n$ with compact support. Let $\Omega_k(\mathbb{R}^n)$ be the space of k-currents — the topological dual of $\Omega_c^k(\mathbb{R}^n)$. Given any k-current $T \in \Omega_k(\mathbb{R}^n)$, the

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+boundary of $T$ is $\partial T \in \Omega_{k-1}(\mathbb{R}^n)$ defined as the adjoint to the exterior derivative $\partial$. A cycle is a current with null boundary.
+
+It is customary to use the *flat topology* on currents [11]. The mass of a k-current $T$ is
+
+$$ (7) \qquad \mathcal{M}(T) = \sup \left\{ T(\omega) : \omega \in \Omega_c^k(\mathbb{R}^n) \text{ and } \sup_{x \in \mathbb{R}^n} |\omega(x)| \le 1 \right\} $$
+
+($|\omega|$ is the usual norm), which generalizes the volume of a submanifold. The *flat norm* of a k-current $T$ is
+
+$$ (8) \qquad |T|_b = \inf\{\mathcal{M}(R) + \mathcal{M}(S) : T = R + \partial S, R \in \mathcal{R}_k(\mathbb{R}^n), S \in \mathcal{R}_{k+1}(\mathbb{R}^n)\}, $$
+
+where $\mathcal{R}_k$ is the space of rectifiable k-currents. The flat norm quantifies the minimal-mass decomposition of a k-current $T$ into a k-current $R$ and the boundary of a $(k+1)$-current $S$.
+
+**Normal cycle:** The normal cycle of a compact definable set $A$ is a definable $(n-1)$-current $N^A$ on the unit sphere cotangent bundle $U^*\mathbb{R}^n \cong S^{n-1} \times \mathbb{R}^n$ that is Legendrian with respect to the canonical 1-form $\alpha$ on $T^*\mathbb{R}^n$. The normal cycle generalizes the unit normal bundle of an embedded submanifold to compact definable sets. The normal cycle is additive: for $A$ and $B$ compact and definable,
+
+$$ (9) \qquad N^{A\cup B} + N^{A\cap B} = N^A + N^B. $$
+
+The intrinsic volume $\mu_k$ is representable as integration of a particular (non-unique) form $\alpha_k \in \Omega^{n-1}U^*\mathbb{R}^n$ against the normal cycle:
+
+$$ (10) \qquad \mu_k(A) = \int_{N^A} \alpha_k. $$
+
+Fu [12] gives a formula for the normal cycle in terms of stratified Morse theory; Nicolaescu [24] gives a nice description of the normal cycle from Morse theory.
+
+**Conormal cycle:** The conormal cycle (also known as the characteristic cycle [17, 20, 26]) of a compact definable set $A$ is a Lagrangian $n$-current $C^A$ on $T^*\mathbb{R}^n$ that generalizes the cone of the unit normal bundle. Indeed, the conormal cycle is the cone over the normal cycle. An intrinsic description for $A$ a submanifold-with-corners is that the conormal cycle is the union of duals to tangent cones at points of $A$. The conormal cycle is additive: for $A$ and $B$ definable,
+
+$$ (11) \qquad C^{A\cup B} + C^{A\cap B} = C^A + C^B. $$
+
+The intrinsic volume $\mu_k$ is representable as integration of a certain (non-unique) form $\omega_k \in \Omega^n T^*\mathbb{R}^n$ (supported by a bounded neighborhood of the zero section of the cotangent bundle) against the conormal cycle:
+
+$$ (12) \qquad \mu_k(A) = \int_{C^A} \omega_k. $$
+
+As the conormal cycles are cones, one can always rescale the forms $\omega_k$ so that they are supported in a given neighborhood of the zero section of the cotangent bundle. We fix the neighborhood once and for all, and will assume henceforth that all $\omega_k$ are supported in the unit ball bundle $B_1^*(\mathbb{R}^n) := \{|P| \le 1\} \subset T^*\mathbb{R}^n$.
+
+The microlocal index theorem [20, 26] gives an interpretation of the conormal cycle in terms of stratified Morse theory.

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+This is to say that if (V,W) is a real solution of matrix equation (18), then $(-Q_n^{-1}VQ_p, -Q_r^{-1}WQ_p)$ is also a real solution of matrix equation (18). Similarly, we can prove that $(-S_n^{-1}VS_p, -S_r^{-1}WS_p)$ is also a real solution of quaternion matrix equation (18). In this case, the conclusion can be obtained along the line of the proof of Theorem 4.2 in [13]. $\square$
+
+**Theorem 8.** Given the quaternion matrices $A \in Q^{n \times n}$, $B \in Q^{p \times p}$, and $C \in Q^{n \times r}$, let
+
+$$ 
+\begin{align}
+f_{(I,A_\sigma)}(s) &= \det(I_{4n} - sA_\sigma) = \sum_{k=0}^{2n} a_{2k}s^{2k}, \tag{29} \\
+p_{A_\sigma}(s) &= \sum_{k=0}^{2n} a_{2k}s^k.
+\end{align}
+$$
+
+Then the matrices $X \in Q^{n \times p}$, $Y \in Q^{r \times p}$ are given by
+
+$$ 
+\begin{align}
+X &= \sum_{k=0}^{2n-1} \sum_{s=k}^{2n-1} \alpha_{2k} (A\hat{A})^{s-k} CZ(\hat{B}B)^s \nonumber \\
+  &\quad + \sum_{k=0}^{2n-1} \sum_{s=k}^{2n-1} \alpha_{2k} (A\hat{A})^{s-k} A\hat{C}\hat{Z}B(\hat{B}B)^s, \tag{30} \\
+Y &= Z p_{A_\sigma}(\hat{B}B),
+\end{align}
+$$
+
+in which Z is an arbitrary quaternion matrix.
+
+*Proof.* If Yakubovich quaternion *j*-conjugate matrix equation (17) has solution $(X, Y)$, then real representation matrix equation (18) has solution $(V, W) = (X_σ, Y_σ)$ with the free parameter $Z_σ$. By Theorems 2 and 7, we have
+
+$$ 
+\begin{align*}
+X_{\sigma} &= \sum_{k=0}^{2n-1} \sum_{j=0}^{k} \alpha_j A_{\sigma}^{j-k} C_{\sigma} P_r Z_{\sigma} B_{\sigma}^j \\
+&= \sum_{k=0}^{2n-1} \sum_{j=2k}^{4n-1} \alpha_{2k} A_{\sigma}^{j-2k} C_{\sigma} P_r Z_{\sigma} B_{\sigma}^j \\
+&= \sum_{k=0}^{2n-1} \alpha_{2k} \left[ \sum_{s=k}^{2n-1} A_{\sigma}^{2s-2k} C_{\sigma} P_r Z_{\sigma} B_{\sigma}^{2s} \right. \\
+&\qquad \left. + \sum_{s=k}^{2n-1} A_{\sigma}^{2s-2k+1} C_{\sigma} P_r Z_{\sigma} B_{\sigma}^{2s+1} \right] \\
+&= \sum_{k=0}^{2n-1} \alpha_{2k} \\
+&\quad \times \left[ \sum_{s=k}^{2n-1} \left( (A\hat{A})^{s-k} \right)_{\sigma} P_n C_{\sigma} P_r Z_{\sigma} \left( (\hat{B}B)^s \right)_{\sigma} P_p \right. \\
+&\qquad \left. + \sum_{s=k}^{2n-1} \left( (A\hat{A})^{s-k} \right)_{\sigma} P_n A_{\sigma} C_{\sigma} P_r Z_{\sigma} B_{\sigma} \left( (\hat{B}B)^s \right)_{\sigma} P_p \right]
+\end{align*}
+$$
+
+$$ 
+= \sum_{k=0}^{2n-1} \alpha_{2k} CZ(\hat{B}\hat{B})^s
+$$
+
+$$ 
++ \sum_{s=k}^{2n-1} ((A\hat{A})^{s-k}) A\hat{C}\hat{Z}\hat{B}(B\hat{B})^s
+$$
+
+(31)
+
+In addition, by Proposition 5, $f_{(I,A_σ)}(s)$ is a real polynomial and $f_{(I,A_σ)}(B_σ) = (p_{A_σ}(B\hat{B}))_σP_p$. So according to Proposition 3, we obtain
+
+$$ 
+Y_σ = Z_σ f_{(I,A_σ)}(B_σ) = Z_σ (p_{A_σ}(B\hat{B}))_σ P_p = (Z_p p_{A_σ}(B\hat{B}))_σ. 
+$$
+
+Thus, the conclusion above has been proved. $\square$
+
+In the following, we provide an equivalent statement of Theorem 8.
+
+**Theorem 9.** Given quaternion matrices $A \in Q^{n \times n}$, $B \in Q^{p \times p}$, and $C \in Q^{n \times r}$, let
+
+$$ 
+\begin{align}
+f_{(I,A_\sigma)}(s) &= \det(I_{4n} - sA_\sigma) = \sum_{k=0}^{2n} a_{2k}s^{2k}, \tag{33} \\
+p_{A_\sigma}(s) &= \sum_{k=0}^{2n} a_{2k}s^k.
+\end{align}
+$$
+
+Then the matrices $X \in Q^{n \times p}$, $Y \in Q^{r \times p}$ given by (30) have the following equivalent form:
+
+$$ 
+\begin{align}
+X &= Q_c(A\hat{A}, C, 2n) S_r(I, A_\sigma) Q_o(B\hat{B}, Z, 2n) \nonumber \\
+  &\quad + Q_c(A\hat{A}, A\hat{C}, 2n) S_r(I, A_\sigma) Q_o(\hat{B}\hat{B}, \hat{Z}\hat{B}, 2n), \tag{34} \\
+Y &= Z p_{A_\sigma}(\hat{B}\hat{B}),
+\end{align}
+$$
+
+in which Z is an arbitrary quaternion matrix.
+
+*Proof.* By the direct computation, we have
+
+$$ 
+\begin{align}
+& \sum_{k=0}^{2n-1} \sum_{s=k}^{2n-1} \alpha_{2k}(A\hat{A})^{s-k} CZ(\hat{B}\hat{B})^s \\
+&= Q_c(A\hat{A}, C, n) S_r(I, A_\sigma) Q_o(\hat{B}\hat{B}, Z, 2n), 
+\end{align}
+$$
+
+$$ 
++ \sum_{k=0}^{2n-1} \sum_{s=k}^{2n-1} \alpha_{2k}(A\hat{A})^{s-k} A\hat{C}\hat{Z}\hat{B}(B\hat{B})^s
+$$
+
+$$ 
+= Q_c(A\hat{A}, A\hat{C}, 2n) S_r(I, A_\sigma) Q_o(\hat{B}\hat{B}, \hat{Z}\hat{B}, 2n).
+$$
+
+Thus, the first conclusion has been proved. With this the second conclusion is obviously true. $\square$
+
+Finally, we consider the solution to the so-called Kalman-Yakubovich *j*-conjugate quaternion matrix equation
+
+$$ X - A\tilde{x}_B = C. 
+$$
+
+Based on the main result proposed above, we have the following conclusions regarding the matrix equation (36).

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+Liquid crystal elastomers wrinkling
+
+Alain Goriely*
+
+L. Angela Mihai†
+
+* Mathematical Institute, University of Oxford, Oxford, UK
+
+† School of Mathematics, Cardiff University, Cardiff, UK
+
+May 14, 2021
+
+Abstract
+
+When a liquid crystal elastomer layer is bonded to an elastic layer, it creates a bilayer with interesting properties that can be activated by applying traction at the boundaries or by optothermal stimulation. Here, we examine wrinkling responses in three-dimensional nonlinear systems containing a monodomain liquid crystal elastomer layer and a homogeneous isotropic incompressible hyperelastic layer, such that one layer is thin compared to the other. The wrinkling is caused by a combination of mechanical forces and external stimuli. To illustrate the general theory, which is valid for a range of bilayer systems and deformations, we assume that the nematic director is uniformly aligned parallel to the interface between the two layers, and that biaxial forces act either parallel or perpendicular to the director. We then perform a linear stability analysis and determine the critical wave number and stretch ratio for the onset of wrinkling. In addition, we demonstrate that a plate model for the thin layer is also applicable when this is much stiffer than the substrate.
+
+**Key words:** liquid crystal elastomers, finite deformation, instabilities, wrinkling, bilayer system, thin film.
+
+# 1 Introduction
+
+Liquid crystal elastomers (LCEs) are complex materials that combine the elasticity of polymeric solids with the self-organisation of liquid crystalline structures [31, 39]. Due to their molecular architecture, consisting of cross-linked networks of polymeric chains containing liquid crystal mesogens, these materials are capable of interesting mechanical responses, including relatively large nonlinear deformations which may arise spontaneously and reversibly under certain external stimuli (e.g., heat, solvents, electric or magnetic field). These qualities suggest many avenues for technological applications, such as soft actuators and soft tissue engineering, but more research efforts are needed before they can be exploited on an industrial scale [32, 49, 54, 60, 67, 81, 84, 86, 92–94].
+
+In particular, the occurrence of material microstructures and the formation of wrinkles in response to external forces or optothermal stimulation are important phenomena that require further investigation. A survey on surface wrinkling in various materials is provided in [58]. The utility of surface instabilities in measuring material properties of thin films is reviewed in [23]. Within the framework of elasticity, relevant experimental and theoretical studies of wrinkling responses in LCEs are as follows:
+
+(I) *Wrinkling and shear stripes formation in liquid crystal elastomers:*
+
+(I.1) *Experimental results.* In [40, 55, 76, 101], stretched LCE samples were reported to display director re-orientation and microstructural shear stripes without wrinkling near the clamped ends, unlike in purely elastic samples. In [1, 2], spontaneous formation and reorientation of surface wrinkles under temperature changes were observed for stiff isotropic elastic plates bonded to a monodomain nematic elastomer foundation. In this case, wrinkles formed either parallel or perpendicular to the nematic director, depending on the temperature at