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, consisting in protein concentration measurements irregularly spread in space and time. 4.1 Choice of measures We will now specify our choice of prior measure [MATH] with justifications: |
[MATH] . Concerning the decay parameter, the only constraint given in the problem so far is positivity and Lebesgue density. However, the diffusion must also satisfy an integrability condition from theorem 2.2 , which is clearly the case for uniform distributions. Finally, we choose [MATH] and [MATH] , maximum paramete... |
[MATH] . The prior measure [MATH] will be chosen as a centred Gaussian measure on [MATH] , with covariance operator on [MATH] [EQUATION] |
where [MATH] is a smoothness parameter tuned to ensure almost-surely continuity of the samples. The precise eigen-decomposition is given as follows [MATH] |
[EQUATION] With these choices done, the Radon-Nikodym density of the prior distribution w.r.t. the reference measure is [EQUATION] |
which is locally Lipschitz (mean-value theorem). We now tune the parameters of [MATH] , by simply choosing [MATH] [MATH] [MATH] and [MATH] (minimizing Kullback-Leibler divergence). Finally, we are capable of specifying the exact form of the generalized Onsager-Mashlup functional using theorem 2.3 |
[EQUATION] In particular, parameters [MATH] and [MATH] are only influenced by their range and contribution to the likelihood (the uniform prior is non-informative), contrary to the source [MATH] |
4.2 Prior and solution map discretization The analysis conducted in all previous sections happens to be valid for infinite dimensional quantities. In practice however, one needs to discretize for numerical experiments. In this work, the solution map is approximated using finite elements in space (FEniCS library in Pyth... |
and ) and finite differences in time. We use 100 basis functions and 30 time steps on a desktop computer (Intel i7-3770 with 8Gb of RAM memory) . We set [MATH] and [MATH] is the final time. Concerning the prior measure, we use a truncated Karhunen-Loeve basis of [MATH] to simulate from it |
[EQUATION] where [MATH] are i.i.d. [MATH] random variables. We consider [MATH] (100 basis functions) thus [MATH] is of dimension 102. All quantities related to negative log-likelihood derivatives (Gradient and Gauss-Newton Hessian matrix) are numerically computed using discrete adjoint methods (see |
or ) to keep scalability in [MATH] . The initial point in the chain is chosen at the MAP location, obtained by minimization of the functional in equation 4.1 (Prior based initialization results in long burnin phase). Practical optimization is done using L-BFGS-B algorithm from the Scipy library |
4.3 Results We now turn to our main objective, the inversion and uncertainty quantification of gap-gene protein concentration from |
. The dataset consists of 508 different measures which are non-uniformly spread in time and space (precise repartition can be seen in figure ). The noise variance parameter is estimated using the following routine (10 iterations, 3 multi-start each): |
1. Find [MATH] minimizing [MATH] using the current noise level [MATH] 2. Update the current variance estimation [MATH] 3. Go back to [MATH] |
The resulting estimated noise level is [MATH] . With this estimated value, we compute our initial MAP estimate (numerical minimization of the Onsager-Machlup functional) and use it as initial point in the MCMC sampling. The Markov chain is ran for [MATH] total iterations and the resulting traceplot is given in figure f... |
The first thousand iterations are used as burnin and according to the autocorrelation function (figure ), we choose to keep one iteration out of a hundred as posterior sample (thinning). |
From this, we compute both posterior mean and MAP estimates, the precise values of decay, diffusion, negative log-likelihood and Onsager-Machlup functional being given in table |
Additionally to the estimated values, one can also look at the marginal distribution on figure Concerning the MAP estimator (figure ), we recover both events described in |
and , that is 2 pikes of protein concentrations. The first happens on the anterior part of the embryo in the early experiment [MATH] . The second is much more intense and happens in the posterior part during the second half of the experiment. The estimated source explains these with an intense and localized increase in... |
Conclusion In this work, we applied the Bayesian inverse problem methodology from to a practical Biological dynamical system. Doing so, we provide a rigorous and detailed analysis of the forward model, existence and continuity of the posterior measure, characterization of maximum a posteriori (MAP) estimates and state-... |
Aknowledgments This work has been supported by Colciencias and ECOS-Nord under the grant C15M04. The authors would like to thank Dr. Xavier Bay for his precious advice and proof reading. |
Appendix A Proofs Proof of theorem 2.1 Using standard notations in PDE theory, let [MATH] [MATH] [MATH] [MATH] [MATH] and [EQUATION] |
equipped with the norm [MATH] then the weak form associated to equation has the following reduced form: [EQUATION] with [EQUATION] |
Existence of a solution map . The PDE operator is uniformly parabolic whenever [MATH] , thus there exists a unique weak solution of equation in [MATH] for every [MATH] (see chapter 7 in |
for instance). Moreover, we have the following energy estimate: [EQUATION] with [MATH] a constant independent of [MATH] Second order Fréchet differentiability of the solution map . Let [MATH] such that [MATH] , and [MATH] then [MATH] |
[EQUATION] with [EQUATION] with [MATH] an other constant independent from [MATH] and: [EQUATION] Moreover, we have: [EQUATION] thus [MATH] is bounded, which shows that [MATH] is Fréchet-differentiable on [MATH] . Consider [MATH] the partial derivative of [MATH] w.r.t. its first variable: |
[EQUATION] which defines a unique solution [MATH] whenever [MATH] (using same arguments than previously). Here, [MATH] is clearly bounded. Because [MATH] is differentiable and [MATH] exists and is bounded, the implicit function theorem applies and [MATH] is differentiable on [MATH] . The second order differentiability ... |
Local Lipschitz continuity of the solution map . Let [MATH] [MATH] such that [MATH] and [MATH] . There exists two unique solution with respect to [MATH] satisfying [MATH] [MATH] |
[EQUATION] which leads to [MATH] and almost-every [MATH] [EQUATION] and equivalently: [EQUATION] Now, letting [MATH] and using the coercivity of [MATH] we get: |
[EQUATION] Now, we use the identity [MATH] and drop the positive term in the left hand side to get [EQUATION] We integrate between [MATH] and [MATH] , use Cauchy-Schwartz and Poincarré’s inequalities and [MATH] to obtain the following estimation |
[EQUATION] The same reasoning applies now to equation 10 to get [EQUATION] We start back from equation to get: [EQUATION] and taking the supremum on the unit ball of [MATH] before integrating: |
[EQUATION] which completes the proof. Proof of theorem 2.2 Following the program in , let us first show that [MATH] exists and is unique and then the continuity of the posterior with respect to the Hellinger distance. |
(Existence and unicity of [MATH] ). Let [MATH] be a probability measure defined as in equation and consider the following Gaussian negative log-likelihood: |
[EQUATION] Since [MATH] is differentiable (theorem 2.1 ), [MATH] is measurable w.r.t. [MATH] for all [MATH] . Consider now the following integral |
[EQUATION] The negative log-likelihood being positive, the integral is finite. Furthermore, we have [MATH] thus [MATH] for every [MATH] . In other words, the following function |
[EQUATION] defines a probability density function w.r.t. [MATH] , and the associated measure is [MATH] , the (unique) posterior distribution of [MATH] |
(Continuity in [MATH] ). It remains to show that the posterior distribution is continuous in [MATH] w.r.t. the Hellinger distance. Following the lines of |
we proove the following sufficient condition [EQUATION] with [MATH] [EQUATION] Now, using continuous injection between spaces, we have |
[EQUATION] the last inequality is given by the energy estimate of theorem 2.1 . It remains to see that [EQUATION] is in [MATH] , that is |
[EQUATION] The first term in the right hand side product is finite by assumption on [MATH] . The second term is finite as well since |
[EQUATION] which is [MATH] integrable by the assumption on [MATH] . Finally, Fubini’s theorem gives the result on the integrability of [MATH] . Now, it remains to follow the proof from |
as is. Proof of theorem 2.3 The posterior measure [MATH] is absolutely continuous w.r.t. [MATH] with the following Radon-Nikodym density: |
[EQUATION] where [MATH] . Let us now show that [MATH] is locally Lispchitz in its first argument: [EQUATION] and since [MATH] , we conclude that [MATH] and thus [MATH] are locally Lipschitz. Now, we follow the lines of |
it comes: [EQUATION] Now, [EQUATION] and finally [EQUATION] A similar argument leads to [EQUATION] We conclude that [MATH] . For a fixed value [MATH] , this quantity is maximized when [MATH] is a minimizer of [MATH] , the proof is then complete. |
# Source: arxiv 1806.05903 # Title: A characterization of Nichols algebras of diagonal type which are free algebras # Sections: all # Downloaded: 2026-03-03T02:41:39.652628+00:00 |
A characterization of Nichols algebras of diagonal type which are free algebras Abstract. This paper is devoted to explore the freeness of Nichols algebras of diagonal type and to determine the dimension of the kernel of the shuffle map considered as an operator acting on the free algebra. Our proof is based on an ineq... |
Keywords : Nichols algebras, free algebras, shuffle map, Lyndon words The second named author was supported by China Scholarship Council |
1. Introduction Since their introduction in the late 70ies by W. Nichols the theory of Nichols algebras enjoyed increasing interest because of its deep interrelation to different research areas. For an overview we refer to |
The strongest results have been obtained for finite-dimensional Nichols algebras of diagonal type, mainly due to the existence of the root system which was introduced in |
, based on deep results of V. Kharchenko on the structure of certain Hopf algebras A general, very difficult question is, what are the roots and their multiplicities of a given Nichols algebra of diagonal type. In the case of finite-dimensional Nichols algebras the answer is known: The roots are the real roots with res... |
Roots of the form [MATH] and their multiplicities have been determined by the authors in . In this paper we address the question when the multiplicity of a root is smaller than in the tensor algebra. In particular, we provide a criterion to decide whether a given Nichols algebra of diagonal type is a free algebra in te... |
The defining ideal of a Nichols algebra is spanned by the kernels of the braided symmetrizer , which decomposes into a product of shuffle maps. In |
the authors study identities involving shuffle maps. We use these identities to study the freeness of Nichols algebras of diagonal type and to determine the dimension of the kernel of the shuffle map. With our results we relate the freeness of Nichols algebras of diagonal type with braiding matrix [MATH] [MATH] |
for all [MATH] , to solutions of a diophantine equation. In Section we define a family [MATH] of elements in the polynomial ring [MATH] , where |
[MATH] Let now [MATH] be a Nichols algebra of diagonal type of rank [MATH] with braiding matrix [MATH] , where [MATH] is a field. |
Theorem 1.1 (see Theorem 4.3 We have [MATH] if and only if [MATH] for all [MATH] with [MATH] Assume that [MATH] has characteristic [MATH] . If [MATH] for some |
[MATH] , then in Section two numbers [MATH] are defined. Theorem 1.2 (see Theorem 6.2 Assume that [MATH] has characteristic [MATH] Let [MATH] and [MATH] such that |
[MATH] [MATH] , and [MATH] for all [MATH] Then [EQUATION] The organization of the paper is as follows. In Section , we recall some basic notions about Lyndon words and introduce notations. We also recall the inequalities on the number of Lyndon words, which the paper is based on. In Section , we discuss the notion of a... |
The paper was written during the visit of the second named author to Marburg University supported by China Scholarship Council. The second named author thanks the department of FB Mathematik and Informatik of Marburg University for hospitality. |
2. Basic Definitions and properties Throughout this paper we write [MATH] and [MATH] for the set of positive integers and the set of integers, respectively. Let [MATH] |
We start with recalling necklaces and Lyndon words, and collect some notations. Let [MATH] . For any [MATH] we write [MATH] . If additionally [MATH] , then let |
[MATH] be the greatest common divisor of [MATH] and if [MATH] , then let [EQUATION] For any [MATH] let [MATH] and for any [MATH] and any |
[MATH] let [MATH] There is a partial ordering on [MATH] denoted by [MATH] [MATH] if and only if [MATH] for all [MATH] Let [MATH] be a set (called the alphabet) of [MATH] elements denoted by |
[MATH] and let [MATH] and [MATH] be the set of words and non-empty words, respectively, with letters in [MATH] For [MATH] in which [MATH] occurs [MATH] times, [MATH] , we write |
[MATH] and call [MATH] the degree of [MATH] We fix a total order [MATH] on [MATH] . There is a total order [MATH] on [MATH] induced by [MATH] , called the lexicographic order: For [MATH] one lets [MATH] if and only if either [MATH] for some [MATH] or there exist [MATH] and [MATH] such that [MATH] [MATH] |
[MATH] , and [MATH] A word [MATH] is called a necklace if for any decomposition [MATH] with [MATH] [MATH] A word [MATH] is Lyndon if for any decomposition |
[MATH] [MATH] [MATH] For any [MATH] let [MATH] and [MATH] denote the number of necklaces and Lyndon words, respectively, of degree [MATH] |
Remark 2.1 Any Lyndon word is a necklace, and for any necklace [MATH] there is a unique pair [MATH] such that [MATH] is Lyndon and [MATH] . Thus, for any [MATH] |
[EQUATION] Remark 2.2 In and one can find explicit formulas for [MATH] and [MATH] for any [MATH] In particular, [EQUATION] In the remaining part of this section we will introduce and study some polynomials, which are crucial for the paper. For any ring [MATH] and any [MATH] let [MATH] and |
[MATH] for any [MATH] Definition 2.3 For any [MATH] with [MATH] let [MATH] be as follows: (1) If [MATH] , where [MATH] , and [MATH] , let |
(4) If [MATH] , where [MATH] [MATH] , let [EQUATION] (5) If [MATH] , where [MATH] [MATH] , let [EQUATION] (6) If [MATH] , where [MATH] [MATH] , let |
[EQUATION] (7) If [MATH] , where [MATH] [MATH] , let [EQUATION] (8) Otherwise, let [EQUATION] Moreover, let [EQUATION] Remark 2.4 |
Let [MATH] with [MATH] By definition of [MATH] [MATH] is a well-defined non-constant monomial in [MATH] and [MATH] is not a non-trivial power of any other monomial. Moreover, |
[MATH] divides [MATH] . In particular, [MATH] in the last case of Definition 2.3 For any [MATH] let [MATH] denote the [MATH] -th cyclotomic polynomial, that is, the minimal polynomial of any primitive [MATH] -th root of [MATH] |
in the complex numbers. Clearly, [MATH] for any [MATH] Next we describe the irreducible factors of the polynomials [MATH] Lemma 2.5 |
Let [MATH] be a Euclidean domain, let [MATH] be a non-zero vector in [MATH] , and let [MATH] . Then there is a matrix [MATH] with [MATH] as its first row and determinant [MATH] |
Proof. View [MATH] as a [MATH] -matrix. Choose a composition [MATH] of elementary column transformations which maps [MATH] to the vector [MATH] . Let [MATH] be the diagonal matrix with diagonal entries [MATH] . Then [MATH] satisfies the desired properties. |
Remark 2.6 Lemma 2.5 also holds for principal ideal domains [MATH] On the other hand, let [MATH] for some field [MATH] and let [MATH] . Then [MATH] , but there are no [MATH] |
with [MATH] . Hence Lemma 2.5 does not hold for this [MATH] Lemma 2.7 Let [MATH] be a non-zero vector in [MATH] with [MATH] Then there is a ring automorphism [MATH] of the Laurent polynomial ring |
[MATH] with [MATH] Proof. By Lemma 2.5 there is a matrix [MATH] with [MATH] as its first row and with determinant [MATH] . Then [MATH] for [MATH] |
defines a ring automorphism of [MATH] as desired. Lemma 2.8 For any [MATH] and any [MATH] with [MATH] the polynomial [MATH] is irreducible. In particular, [MATH] |
is the unique factorization of [MATH] into irreducibles, and each irreducible factor of [MATH] is of the form [MATH] for some [MATH] |
Proof. Let [MATH] and [MATH] with [MATH] For any [MATH] let [EQUATION] Then [MATH] and [MATH] by construction. By Lemma 2.7 there is a ring automorphism [MATH] of the Laurent polynomial ring |
[MATH] with [MATH] Thus [MATH] is irreducible in [MATH] Since [MATH] is not divisible in [MATH] by any [MATH] with [MATH] the polynomial [MATH] is irreducible. |
Lemma 2.9 Let [MATH] with [MATH] Suppose that there exist [MATH] such that [MATH] Then [MATH] and [MATH] are relatively prime if and only if [MATH] In particular, [MATH] and [MATH] |
are relatively prime whenever [MATH] Proof. Recall that [MATH] is not constant. Thus, if [MATH] and [MATH] are relatively prime, then [MATH] |
Conversely, suppose that [MATH] and [MATH] are not relatively prime. Then, by Remark 2.4 [MATH] and [MATH] are not relatively prime. Let [MATH] be a non-constant common factor of [MATH] and [MATH] Lemma 2.8 implies that there exist non-constant monic polynomials [MATH] with [MATH] In particular, [MATH] Let [MATH] with ... |
[EQUATION] From Equation ( ) and ( ), one gets [EQUATION] Similarly, using Equation ( ) and ( ), one gets [EQUATION] Thus [MATH] . Let [MATH] . Replacing [MATH] with [MATH] and |
[MATH] with [MATH] in Equation ( ), we get [MATH] , and hence [MATH] because of [MATH] . Thus [MATH] . It follows that [MATH] Now we pass to another family of polynomials, which are the main reason for our interest in the family [MATH] |
Definition 2.10 For any [MATH] [MATH] , let [EQUATION] Note that any [MATH] with [MATH] divides [MATH] and any [MATH] [MATH] , and hence [MATH] . Similarly, [MATH] implies that |
[MATH] . Therefore the numerator and the denominator of [MATH] are polynomials. If [MATH] for some [MATH] and [MATH] , then [MATH] and |
[EQUATION] In order to show that every other [MATH] is a polynomial, we use some results in about the number of Lyndon words. Using Equation ( ), these can be restated as follows. |
Theorem 2.11 , Lemma 4.1, 4.2, Theorem 1.2] Let [MATH] . Assume that [MATH] for some [MATH] . Then, (1) [EQUATION] (2) [EQUATION] |
if and only if [MATH] is one of the cases [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] in Definition 2.3 different from [MATH] Lemma 2.12 |
Let [MATH] If there exist [MATH] [MATH] , such that [MATH] [MATH] , then [MATH] is a polynomial in [MATH] Proof. The numerator of [MATH] is a multiple of |
[MATH] and the denumerator of [MATH] is a divisor of [MATH] Thus the claim follows from Theorem 2.11 (1). Proposition 2.13 Let [MATH] with [MATH] for some [MATH] . Then [MATH] is the product of the irreducible factors of |
[MATH] Proof. We follow Definition 2.3 case by case to compare [MATH] and [MATH] . Then the claim follows directly from Lemma 2.8 |
(1) If [MATH] for some [MATH] [MATH] , then the assumptions of the lemma are not fulfilled. (2) Assume that [MATH] [MATH] [MATH] [MATH] Then [MATH] [MATH] , and |
[EQUATION] where we used Equations ( ) and ( ). (3) Assume that [MATH] [MATH] [MATH] [MATH] Then [MATH] and [MATH] Moreover, [EQUATION] |
If [MATH] is even, then [MATH] by Theorem 2.11 (2). Thus [EQUATION] by Equation ( ). If [MATH] is odd, then [MATH] by Theorem 2.11 (2). Therefore |
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