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# Real and Complex Analysis

Authors:
... where dµ dμ is the Radon-Nykodin derivative [51]; an example is the KL divergence. ...
... and in ♥ we used the Cauchy-Schwarz inequality [51,Theorem 4.2]. Since E µ [ w, x ] is 0-convex (by Proposition 2.12), −J is (−2ρ w 2 )-convex along any interpolating curve. ...
... where • in △ we use Cauchy-Schwarz inequality [51,Theorem 4.2]; ...
Preprint
We study first-order optimality conditions for constrained optimization in the Wasserstein space, whereby one seeks to minimize a real-valued function over the space of probability measures endowed with the Wasserstein distance. Our analysis combines recent insights on the geometry and the differential structure of the Wasserstein space with more classical calculus of variations. We show that simple rationales such as "setting the derivative to zero" and "gradients are aligned at optimality" carry over to the Wasserstein space. We deploy our tools to study and solve optimization problems in the setting of distributionally robust optimization and statistical inference. The generality of our methodology allows us to naturally deal with functionals, such as mean-variance, Kullback-Leibler divergence, and Wasserstein distance, which are traditionally difficult to study in a unified framework.
... Let us introduce the following Möbius transformation (see [25]): ...
... , x n ). [25], we see that ...
... By Rouché's theorem [25], for every m ≥ N , F m also has exactly one root in the ball B(z (∞) , ǫ). By Theorem 3.2, it is z (m) . ...
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This paper gives a new approach for the maximum likelihood estimation of the joint of the location and scale of the Cauchy distribution. We regard the joint as a single complex parameter and derive a new form of the likelihood equation of a complex variable. Based on the equation, we provide a new iterative scheme approximating the max- imum likelihood estimate. We also handle the equation in an algebraic manner and derive a polynomial containing the maximum likelihood estimate as a root. This algebraic approach provides another scheme approximating the maximum likelihood estimate by root-finding algorithms for polynomials, and furthermore, gives non-existence of closed- form formulae for the case that the sample size is five. We finally provide some numerical examples to show our method is effective.
... Let Ω be a metric space; let M be a finite Borel measure over Ω. It follows from the well-known version of Lusin's Theorem (e.g., Theorem 2.24 in Rudin [6]) that if Ω is locally compact, if M is Radon, and if : Ω → R is Borel-measurable, then for every > 0 there is some bounded, uniformly continuous function : Ω → R such that the set of points at which and possibly disagree has measure less than . For our purposes, we refer to a Borel function : Ω → R satisfying the conclusion of the above proposition as (M)-almost uniformly continuous, where Ω is simply a metric space and M, with respect to which the meaning of "almost" is clearly assigned, is simply a finite Borel measure over Ω. ...
... We will stick to the standard measure-theoretic definitions of outer regularity (approximation by open sets from without) and inner regularity (approximation by compact sets from within) of measures (Rudin [6] or Folland [3], for concreteness). For our purposes, it would be convenient to introduce the following terminology. ...
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With a simple short proof, this article improves a classical approximation result of Lusin's type; specifically, it is shown that, on any given finite Borel measure space with the ambient space being a Polish metric space, every Borel real-valued function is almost a bounded, uniformly continuous function in the sense that for every $\eps > 0$ there is some bounded, uniformly continuous function such that the set of points at which they would not agree has measure less than $\eps$. This result also complements the known result of almost uniform continuity of Borel real-valued functions on a finite Radon measure space whose ambient space is a locally compact metric space.
... The zeta function is extracted at the point of collapse of the drop where it possesses a high probability to exhibit the property of a surge. Generally, continuous functions are a set of mathematical functions classically defined in a variety of ways [1]- [5]. They are seen to possess various useful properties which can be utilized in many applications including modeling. ...
... From the plots generated for the synthesized function and the tabulated values, it can be seen that the proposed function corresponds to a dying-surge. Therefore, the dying-surge is modeled as a continuous function as per eqn (5). ...
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Riemann zeta is defined as a function of a complex variable that analytically continues the sum of the Dirichlet series, when the real part is greater than unity. In this paper, the Riemann zeta associated with the finite energy possessed by a 2mm radius, free falling water droplet, crashing into a plane is considered. A modified zeta function is proposed which is incorporated to the spherical coordinates and real analysis has been performed. Through real analytic continuation, the single point of contact of the drop at the instant of touching the plane is analyzed. The zeta function is extracted at the point of destruction of the drop, where it defines a unique real function. A special property is assumed for some continuous functions, where the function’s first derivative and first integral combine together to a nullity at all points. Approximate reverse synthesis of such a function resulted in a special waveform named the dying-surge. Extending the proposed concept to general continuous real functions resulted in the synthesis of the corresponding function’s Dying-surge model. The Riemann zeta function associated with the water droplet can also be modeled as a dying–surge. The Dying- surge model corresponds to an electrical squeezing or compression of a waveform, which was originally defined over infinite arguments, squeezed to a finite number of values for arguments placed very close together with defined final and penultimate values. Synthesized results using simulation software are also presented, along with the analysis. The presence of surges in electrical circuits will correspond to electrical compression of some unknown continuous, real current or voltage function and the method can be used to estimate the original unknown function.
... Since t → P(X t ∈ A) is bounded and as Fubini's theorem implies that t → A p(t, x) dx belongs to L 1 ([0, T ]), it follows from Lebesgue differentiation theorem (see Theorem 7.7 in [Rud87]) that for almost all s ∈ [0, T ] ...
... where λ denotes the Lebesgue measure on R d . From Lebesgue differentiation theorem (see Theorem 7.7 in [Rud87]), we deduce that for all x ∈ E,ũ(x, ·) = u(x) λ-almost everywhere. We prove that for all n ≥ 1, u n is continuous. ...
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We prove It{\^o}'s formula for the flow of measures associated with an It{\^o} process having a bounded drift and a uniformly elliptic and bounded diffusion matrix, and for functions in an appropriate Sobolev-type space. This formula is the almost analogue, in the measure-dependent case, of the It{\^o}-Krylov formula for functions in a Sobolev space on $R^+ \times R^d$.
... The function (14) has a unique local solution ...
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In this paper, we study solutions and drift homogenization for a class of viscous lake equations using the method of semigroups of bounded operators. Suppose that the initial value (t0,u0) ∈ U, i.e., u0 = u(t0), for some Hölder continuous function u on [0,T] with smooth function value u(t) ∈ DL2 (Ω)1/2 satisfying ∂jui = 0(i ≠ j) and b(x) ∈ C∞ (Ω). Then, the initial value problem for viscous lake equations has a unique smooth local strong solution. Using this result, we study the drift homogenization for the three-dimensional stationary Stokes equation in the usual sense, bDL2(Ω). Mathematics Subject Classification (2010). Primary 35Q30, 35B27; Secondary 76N10, 76M50, 47D06.
... However, the flexibility Σ ∈ S ++ N (i.e., considering correlation rather than coherence coefficients) is required in order to conduct proper mathematical derivations, and to ensure the algorithmic stability of the maximum likelihood estimator algorithms: there is, to the best of our knowledge, no explicit (nor tractable) maximum likelohood estimator for the modulus-argument decomposition when assuming some additional phase and/or low-rank structure in the covariance matrix. This is because the modulus and argument are not holomorphic functions [17,18]. This also probably explains why most related works rely on plug-in estimates of the matrix Υ, as observed in [9] (which also implicitly reformulates phase-linking using the real core parameterization). ...
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Phase linking is a prominent methodology to estimate coherence and phase difference in interferometric synthetic-aperture radar. This method is driven by a maximum likelihood estimation approach, which allows to fully exploit all the possible interferograms from a time series. Its performance is however known to be affected by the accuracy of the covariance matrix estimation step, which usually requires to introduce additional prior information on its structure when there is a small sample support (spatial window). Moreover, most phase linking algorithms are built upon the sample covariance matrix, due to the assumption of an underlying Gaussian distribution. In a scenario where SAR data is high resolution, or when the study area is spatially heterogeneous (e.g., urban area), this assumption can also limit the accuracy of the covariance matrix estimation step. Considering the two aforementioned issues, we introduce alternative statistical models, whose maximum likelihood estimators then yield new phase linking algorithms. In order to be robust to non-Gaussian data, we consider the use of a more general model of scaled mixture of Gaussian. To address small sample support issues, we also generalize this approach to a possibly low-rank structured covariance matrix. A unified algorithm to perform phase linking given these models is then derived and validated by simulations and a real data case (Sentinel-1 data). Index Terms-interferometric synthetic aperture radar (In-SAR), distributed scatterers (DS), phase linking (PL), maximum likelihood estimator (MLE), Scaled Gaussian distribution, covari-ance matrix, low-rank (LR).
... For more details on the foundations of signal processing we refer the reader to [96] or [116], and elements of function analysis can be found in [104]. ...
Thesis
Cryptography on embedded devices can be exposed by side-channel attacks that extract sensitive information during the algorithm execution. This leak is usually measured in the electrical consumption signal or the scattered electromagnetic field. As of today, current methods are based on the signal in the temporal domain. But as the development of new countermeasures progress, signals becomes more and more complex, noisy and desynchronized. The analysis in a time-frequency space allows us to identify these countermeasures. In particular, wavelet transforms provide detailed time-frequency (time-scale) representations that highlight the different events coming from the execution of a cryptographic algorithm. The goal of this thesis is to study the use of wavelet transform for developing new methods for the analysis of side-channel signals.
... For s 01 , we have ∆ 01 (T ) ∼ {|dw t |>δ max } Θ(e −β(T −s) )(dw t − δ max )dt. (17) As the benign observation dw t follows a light-tailed distribution and δ max is chosen as the 95% quantile of the normal observation distribution, the above integral would mainly happens for a discrete set of frames when the dynamic model underlying MOT characterizes less accurately about the target object (i.e., {|dw t | > δ max }), which vanishes because the set of {|dw t | > δ max } has a zero measure [55]. ...
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Self-driving cars (SDC) commonly implement the perception pipeline to detect the surrounding obstacles and track their moving trajectories, which lays the ground for the subsequent driving decision making process. Although the security of obstacle detection in SDC is intensively studied, not until very recently the attackers start to exploit the vulnerability of the tracking module. Compared with solely attacking the object detectors, this new attack strategy influences the driving decision more effectively with less attack budgets. However, little is known on whether the revealed vulnerability remains effective in end-to-end self-driving systems and, if so, how to mitigate the threat. In this paper, we present the first systematic research on the security of object tracking in SDC. Through a comprehensive case study on the full perception pipeline of a popular open-sourced self-driving system, Baidu's Apollo, we prove the mainstream multi-object tracker (MOT) based on Kalman Filter (KF) is unsafe even with an enabled multi-sensor fusion mechanism. Our root cause analysis reveals, the vulnerability is innate to the design of KF-based MOT, which shall error-handle the prediction results from the object detectors yet the adopted KF algorithm is prone to trust the observation more when its deviation from the prediction is larger. To address this design flaw, we propose a simple yet effective security patch for KF-based MOT, the core of which is an adaptive strategy to balance the focus of KF on observations and predictions according to the anomaly index of the observation-prediction deviation, and has certified effectiveness against a generalized hijacking attack model. Extensive evaluation on $4$ KF-based existing MOT implementations (including 2D and 3D, academic and Apollo ones) validate the defense effectiveness and the trivial performance overhead of our approach.
... Phragmén-Lindelöf theorems (cf. [2,40,56,60]) are powerful complex analysis tools that extend the maximum modulus principle to certain unbounded domains. They are often used in the presence of exponential bounds, which are, for example, available in the analysis of heat kernels of elliptic second-order differential operators. ...
Article
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We consider fractional Schrödinger operators with possibly singular potentials and derive certain spatially averaged estimates for its complex-time heat kernel. The main tool is a Phragmén–Lindelöf theorem for polynomially bounded functions on the right complex half-plane. The interpolation leads to possibly nonoptimal off-diagonal bounds.
... Let R ⊂ Ω be a small parallelepiped, and let τ be a translation such that τ (R) ⊂ Ω. We will show that π # µ(R) = π # µ(τ (R)), and since this will be true for all R and all τ , π # µ must be a positive multiple of Lebesgue on Ω [18,Thm. 2.20]. ...
Preprint
Recent works have proposed linear programming relaxations of variational optimization problems subject to PDE constraints based on the occupation measure formalism. The main appeal of these methods is the fact that they rely on convex optimization, typically semidefinite programming. In this work we close an open question related to this approach. We prove that the classical and relaxed minima coincide when the codimension equals one, complementing existing results in dimension one. In order to do so, we prove a generalization of the Hardt-Pitts decomposition of normal currents applicable in our setting. We also show by means of a counterexample that if both the dimension and codimension are greater than one there may be a positive gap. The example we construct to show the latter serves also to show that sometimes occupation measures may represent a more conceptually-satisfactory solution'' than their classical counterpart, so that -- even though they may not be equivalent -- algorithms rendering accessible the minimum in the larger space of occupation measures remain extremely valuable.
... Moreover, we know that Φ t (x) = A t x → 0, for all x ∈ R d , since all eigenvalues of A are strictly inside the unit circle. From the continuity of φ, we have that φ(Φ t (x)) → φ(0), and since φ is bounded, from the Dominated Convergence Theorem [72], we have that ...
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Uncertainty propagation has established itself as a fundamental area of research in all fields of science and engineering. Among its central topics stands the problem of modeling and propagating distributional uncertainty, i.e., the uncertainty about probability distributions. In this paper, we employ tools from Optimal Transport to capture distributional uncertainty via Optimal Transport ambiguity sets, which we show to be very natural and expressive, and to enjoy powerful topological, geometrical, computational, and statistical features and guarantees. We show that these ambiguity sets propagate nicely and intuitively through nonlinear, possibly corrupted by noise, transformations, and that in many cases the result of the propagation is again an Optimal Transport ambiguity set. Moreover, whenever this is not the case, we show that the result of the propagation can be tightly upper bounded by another Optimal Transport ambiguity set. Importantly, our methodology allows us to capture exactly how complex systems shape distributional uncertainty. To conclude, we exemplify our findings in three fundamental applications in forward propagation, inverse problems, and distributionally robust optimization.
... We are going to apply the Hadamard three line theorem (see [Rud87,Theorem 12.8]) to the holomorphic family of distributions u(z). From (2.19), we have I g f ∈ L 2 (∂ − M, µ ∂ ), but we can also write the pointwise bound ...
Preprint
The lens data of a Riemannian manifold with boundary is the collection of lengths of geodesics with endpoints on the boundary together with their incoming and outgoing vectors. We show that negatively-curved Riemannian manifolds with strictly convex boundary are locally lens rigid in the following sense: if $g_0$ is such a metric, then any metric $g$ sufficiently close to $g_0$ and with same lens data is isometric to $g_0$, up to a boundary-preserving diffeomorphism. More generally, we consider the same problem for a wider class of metrics with strictly convex boundary, called metrics of Anosov type. We prove that the same rigidity result holds within that class in dimension $2$ and in any dimension, further assuming that the curvature is non-positive.
... Normalizing flows model x as a transformation T of a real vector z ∈ R d sampled from the chosen base distribution p z (z), which could be as simple as a multivariate normal distribution. With invertible and differentiable T (and hence T −1 ) and using the change of variable formula [31], we obtain the density of x as: where J T is the Jacobian of T . Since z = T −1 (x), p x (x) can also be written in terms of x and the Jacobian of T −1 : ...
Preprint
To represent people in mixed reality applications for collaboration and communication, we need to generate realistic and faithful avatar poses. However, the signal streams that can be applied for this task from head-mounted devices (HMDs) are typically limited to head pose and hand pose estimates. While these signals are valuable, they are an incomplete representation of the human body, making it challenging to generate a faithful full-body avatar. We address this challenge by developing a flow-based generative model of the 3D human body from sparse observations, wherein we learn not only a conditional distribution of 3D human pose, but also a probabilistic mapping from observations to the latent space from which we can generate a plausible pose along with uncertainty estimates for the joints. We show that our approach is not only a strong predictive model, but can also act as an efficient pose prior in different optimization settings where a good initial latent code plays a major role.
... Hence these functions also can be characterized in terms of an integral representation, however, with complex measures. Recall that complex measures by definition are finite, see e.g., [81,Chapter 6], and hence the representation of the form of Equation (2) in Part I is used. Considering the difference of the two functions in the example above shows that there are non-trivial quasi-Herglotz functions vanishing identically in one half-plane. ...
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Part II of the review article focuses on the applications of Herglotz-Nevanlinna functions in material sciences. It presents a diverse set of applications with details and the role of Herglotz-Nevanlinna functions clearly pointed out. This paper is concluded by a collection of existent generalizations of the class of Herglotz-Nevanlinna functions that are motivated by potential applications.
... For notation and main definitions, see [33]. where S : D −→ R is an arbitrary function, S(D) is bounded and (for a fixed ρ): U ρ (x, y) = (u, v) ∈ R 2 : (u − x) 2 + (v − y) 2 ≤ ρ 2 , ...
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We present a simple mathematical model for the mammalian low visual pathway, taking into account its key elements: retina, lateral geniculate nucleus (LGN), primary visual cortex (V1). The analogies between the cortical level of the visual system and the structure of popular CNNs, used in image classification tasks, suggests the introduction of an additional preliminary convolutional module inspired to precortical neuronal circuits to improve robustness with respect to global light intensity and contrast variations in the input images. We validate our hypothesis on the popular databases MNIST, FashionMNIST and SVHN, obtaining significantly more robust CNNs with respect to these variations, once such extra module is added.
... See[21] for the background. The inequality "≥" follows from| f dφ| ≤ |f | d|φ| ≤ f ∞ d|φ| = f ∞ φ 1where |φ| is the total variation measure; the inequality "≤" is in the Riesz Representation Theorem.9 ...
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Variational (or, parameterized) quantum circuits are quantum circuits that contain real-number parameters, that need to be optimized/"trained" in order to achieve the desired quantum-computational effect. For that training, analytic derivatives (as opposed to numerical derivation) are useful. Parameter shift rules have received attention as a way to obtain analytic derivatives, via statistical estimators. In this paper, using Fourier Analysis, we characterize the set of all shift rules that realize a given fixed partial derivative of a multi-parameter VQC. Then, using Convex Optimization, we show how the search for the shift rule with smallest standard deviation leads to a primal-dual pair of convex optimization problems. We study these optimization problems theoretically, prove a strong duality theorem, and use it to establish optimal dual solutions for several families of VQCs. This also proves optimality for some known shift rules and answers the odd open question. As a byproduct, we demonstrate how optimal shift rules can be found efficiently computationally, and how the known optimal dual solutions help with that.
... We present here a partial version of this theorem which will be enough for our purposes. We will not give the proof of this result here; the reader may find various proofs in [31], [36], [39], and [68]. In the rest of the text we will use the abbreviation QADC classes for quasianalytic Denjoy-Carleman classes. ...
... En prenant le supremum sur les r < 1 et en appliquant le théorème de convergence monotone, on obtient alors [70,Chapitre 11]). Réciproquement, si on commence avec f ∈ H 2 , alors par le théorème de Fatou, f possède des limites radiales presque partout, et la fonction limite est dans L 2 (T), ses coefficients de Fourier négatifs sont nuls, et f coïncide avec son intégrale de Poisson. ...
Thesis
Cette thèse est consacrée à l’étude des opérateurs de Toeplitz et des opérateursde composition sur les espaces de de Branges-Rovnyak H(b), qui sont une classed’espaces de Hilbert de fonctions analytiques dans le disque unité ouvert D duplan complexe, paramétrée par une fonction b dans la boule unité de H∞. Cesespaces ont été introduits, dans les années 60, pour construire un modèle pourles contractions sur un espace de Hilbert, mais on s’est aperçu depuis qu’ilsavaient un rôle important à jouer dans de nombreuses questions de théorie desopérateurs et de théorie des fonctions d’une variable complexe.Dans cette thèse, nous nous sommes intéressés, d’une part, à l’étude desopérateurs de Toeplitz T¯φ, où φ ∈ H∞, qui agissent de façon borné sur H(b).Nous avons donné quelques estimations de la norme de ces opérateurs, puis nousavons obtenu une caractérisation de la compacité. Nous avons également étudiéla dynamique de ces opérateurs, en donnant une caractérisation de l’hypercyclicitéet en construisant un vecteur cyclique commun. Comme souvent dansla théorie des espaces de de Branges-Rovnyak, ces propriétés vont dépendre dufait que log(1 − |b|) est intégrable ou non sur T. Nous avons ainsi généralisé uncertain nombre de résultats connus pour les opérateurs de Toeplitz standardTφ définis sur H² et les opérateurs de Toeplitz tronqué Aθφ définis sur l’espacemodèle Kθ = H(θ) (correspondant au cas où b = θ est une fonction intérieure).D’autre part, nous nous sommes également intéressés à une autre classed’opérateurs naturels, à savoir les opérateurs de composition sur H(b). Dansle cas, où la fonction b est une fonction rationnelle et telle que log(1 − |b|) estintégrable sur T, nous avons caractérisé la bornitude et la compacité des opérateursde composition Cφ sur H(b), en exploitant un lien intéressant avec lesopérateurs de composition à poids sur H². Nous avons en particulier généraliséplusieurs résultats obtenus précédemment par Sarason-Silva pour les opérateursde composition sur les espaces de Dirichlet locaux.
... In particular, we review Blaschke products and the Helson-Sarason Theorem. We note that the standard presentation of the Blaschke products is motivated by inner functions in Hardy spaces presented over the unit disk whose boundary is the unit circle (Rudin, 1987;Garcia et al., 2016). In our case, we are interested in the right hand half plane whose boundary is the jω-axis. ...
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Numerous physical systems are described by ordinary or partial differential equations whose solutions are given by holomorphic or meromorphic functions in the complex domain. In many cases, only the magnitude of these functions are observed on various points on the purely imaginary jw-axis since coherent measurement of their phases is often expensive. However, it is desirable to retrieve the lost phases from the magnitudes when possible. To this end, we propose a physics-infused deep neural network based on the Blaschke products for phase retrieval. Inspired by the Helson and Sarason Theorem, we recover coefficients of a rational function of Blaschke products using a Blaschke Product Neural Network (BPNN), based upon the magnitude observations as input. The resulting rational function is then used for phase retrieval. We compare the BPNN to conventional deep neural networks (NNs) on several phase retrieval problems, comprising both synthetic and contemporary real-world problems (e.g., metamaterials for which data collection requires substantial expertise and is time consuming). On each phase retrieval problem, we compare against a population of conventional NNs of varying size and hyperparameter settings. Even without any hyper-parameter search, we find that BPNNs consistently outperform the population of optimized NNs in scarce data scenarios, and do so despite being much smaller models. The results can in turn be applied to calculate the refractive index of metamaterials, which is an important problem in emerging areas of material science.
... (6.65) D'après l'inégalité de Jenssen [Rud87], on doit avoir ln P q ≥ ln P q et donc τ ens ≤ τ typ . Ici, ces deux quantités ne sont égales que pour q ∈ {0; 1}, c'est-à-dire lorsque les moments P q sont constants. ...
Thesis
En dimension trois, un système quantique désordonné peut présenter une transition entre un état métallique/diffusif à faible désordre et un état isolant/localisé à fort désordre. Au voisinage de cette transition appelée transition d'Anderson, il est connu que les fonctions d'onde des états propres présentent des fluctuations géantes et un caractère multifractal. Dans ce manuscrit, nous utilisons un système spécifique, le rotateur pulsé --- également appelé kicked rotor --- quasi périodique pour étudier les propriétés de multifractalité de paquets d'onde. C'est un système unidimensionnel, donc facile à étudier expérimentalement et à simuler numériquement, mais sa dépendance temporelle est telle qu'il présente une transition d'Anderson aisément contrôlable. Nous étudions numériquement et interprétons théoriquement les propriétés de multifractalité des paquets d'onde au voisinage de la transition d'Anderson. Nous montrons que celles-ci permettent de remonter partiellement aux propriétés de multifractalité des états propres en dimension trois, ouvrant ainsi des perspectives pour une étude expérimentale future.
... We have considered eight complex planes and three multilevel contours as an example, but one can extend these examples to demonstrate the intersection of many more complex planes and contours passing through them. Although multilevel contours are newly introduced here in this article, the principles associated with contours on a single complex plane can be found in any standard textbooks, see for example [1,2,3,4]. ...
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A new concept called multilevel contours is introduced through this article by the author. Theorems on contours constructed on a bundle of complex planes are stated and proved. Multilevel contours can transport information from one complex plane to another. Within a random environment, the behavior of contours and multilevel contours passing through the bundles of complex planes are studied. Further properties of contours by a removal process of the data are studied. The concept of 'islands' and 'holes' within a bundle is introduced through this article. These all constructions help to understand the dynamics of the set of points of the bundle. Further research on the topics introduced here will be followed up by the author. These include closed approximations of the multilevel contour formations and their removal processes. The ideas and results presented in this article are novel.
... Phragmén-Lindelöf theorems (cf. [2,31,40,43]) are powerful complex analysis tools that extend the maximum modulus principle to certain unbounded domains. They are often used in the presence of exponential bounds, which are, e.g., available in the analysis of heat kernels of elliptic second-order differential operators. ...
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We consider fractional Schrödinger operators with possibly singular potentials and derive certain spatially averaged estimates for its complex-time heat kernel. The main tool is a Phragmén-Lindelöf theorem for polynomially bounded functions on the right complex half-plane. The interpolation leads to possibly nonoptimal off-diagonal bounds.
... In fact, for subcategories of finitely encoded persistence modules, any additive amplitude can be obtained by integrating the Hilbert function with respect to something slightly weaker than a measure, i.e. a content, on the indexing poset. This may be understood as a Riesz-(Markov)-Representation type theorem for additive amplitudes (see [Rud87,Thm. 2.14] for a reference of the latter). ...
Preprint
The use of persistent homology in applications is justified by the validity of certain stability results. At the core of such results is a notion of distance between the invariants that one associates to data sets. While such distances are well-understood in the one-parameter case, the situation for multiparameter persistence modules is more challenging, since there is no generalisation of the barcode. Here we introduce a general framework to study stability questions in multiparameter persistence. We introduce amplitudes -- invariants that arise from assigning a non-negative real number to each persistence module, and which are monotone and subadditive in an appropriate sense -- and then study different ways to associate distances to such invariants. Our framework is very comprehensive, as many different invariants that have been introduced in the Topological Data Analysis literature are examples of amplitudes, and furthermore, many known distances for multiparameter persistence can be shown to be distances from amplitudes. Finally, we show how our framework can be used to prove new stability results.
... Furthermore, it is clear that |f (x)| ≤ M for every x for which this second derivative exists. The result of[Rud87, Theorem 7.20] implies that ...
Thesis
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Sampling from probability distributions is a problem of significant importance in Statistics and Machine Learning. The approaches for the latter can be roughly classified into two main categories, that is the frequentist and the Bayesian. The first is the MLE or ERM which boils down to optimization, while the other requires the integration of the posterior distribution. Approximate sampling methods are hence applied to estimate the integral. In this manuscript, we focus mainly on Langevin sampling which is based on discretizations of Langevin SDEs. The first half of the introductory part presents the general mathematical framework of statistics and optimization, while the rest aims at the historical background and mathematical development of sampling algorithms.The first main contribution provides non-asymptotic bounds on convergence LMC in Wasserstein error. We first prove the bounds for LMC with the time-varying step. Then we establish bounds in the case when the gradient is available with a noise. In the end, we study the convergence of two versions of discretization, when the Hessian of the potential is regular.In the second main contribution, we study the sampling from log-concave (non-strongly) distributions using LMC, KLMC, and KLMC with higher-order discretization. We propose a constant square penalty for the potential function. We then prove non-asymptotic bounds in Wasserstein distances and provide the optimal choice of the penalization parameter. In the end, we highlight the importance of scaling the error for different error measures.The third main contribution focuses on the convergence properties of convex Langevin diffusions. We propose to penalize the drift with a linear term that vanishes over time. Explicit bounds on the convergence error in Wasserstein distance are proposed for the PenalizedLangevin Dynamics and Penalized Kinetic Langevin Dynamics. Also, similar bounds are proved for the Gradient Flow of convex functions.
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This paper is concerned with the exponential stability of the stochastic complex networks with Markovian switching topologies. It is worth emphasizing that the topological structure of the stochastic control system is Markovian switching and it is neither required that all switching subnetworks contain a spanning tree nor that they are strongly connected. Moreover, a new type of control, aperiodically intermittent discrete-time state observations control, is proposed. By using the Lyapunov method and graph theory, a theorem with the sufficient conditions for exponential convergence of the state to zero and two corollaries are established. In addition, our theoretical results are used to discuss exponential stability of stochastic coupled oscillators and a communication network model, respectively. Finally, two numerical examples are given to verify the effectiveness of the results.
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Gradient descent (GD) type optimization schemes are the standard methods to train artificial neural networks (ANNs) with rectified linear unit (ReLU) activation. Such schemes can be considered as discretizations of gradient flows (GFs) associated to the training of ANNs with ReLU activation and most of the key difficulties in the mathematical convergence analysis of GD type optimization schemes in the training of ANNs with ReLU activation seem to be already present in the dynamics of the corresponding GF differential equations. It is the key subject of this work to analyze such GF differential equations in the training of ANNs with ReLU activation and three layers (one input layer, one hidden layer, and one output layer). In particular, in this article we prove in the case where the target function is possibly multi-dimensional and continuous and in the case where the probability distribution of the input data is absolutely continuous with respect to the Lebesgue measure that the risk of every bounded GF trajectory converges to the risk of a critical point. In addition, in this article we show in the case of a 1-dimensional affine linear target function and in the case where the probability distribution of the input data coincides with the standard uniform distribution that the risk of every bounded GF trajectory converges to zero if the initial risk is sufficiently small. Finally, in the special situation where there is only one neuron on the hidden layer (1-dimensional hidden layer) we strengthen the above named result for affine linear target functions by proving that that the risk of every (not necessarily bounded) GF trajectory converges to zero if the initial risk is sufficiently small.
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