Math/Stat Colloquium

3:30 p.m., August 29, 2024

TLC 029

Ingmar A. Saberi (Ludwig-Maximilians-Universitat Munchen, Munich, Germany)

Lie algebras, homotopy generalizations, and the calculus of variations

Abstract:  I'll give a friendly, somewhat unconventional overview of some important ideas in the modern approach to classical and quantum field theories, centered around the notion of a Lie algebra and a homotopy generalization called an L-infinity algebra.  Both the partial differential equations defining a classical field theory and any collection of infinitesimal symmetries acting locally on it can be captured by structures of this kind.  Time permitting, I'd like to touch on some results that use this approach to uncover new symmetries in higher-dimensional field theories.  No prior exposure to any of these topics will be assumed; one chief aim will be to give as approachable an introduction as possible to some of the mathematical structures that appear naturally in theoretical physics.

3:30 p.m., September 19, 2024

TLC 029

Alex Vakanski, University of Idaho, Dep't of Nuclear Engineering & Industrial Management (Idaho Falls)

Enhancing Adversarial Robustness of Machine Learning Models for Medical Image Classification

Abstract: With the widespread adoption of machine learning-based approaches in various application domains, understanding their vulnerabilities to adversarial attacks is crucial to ensuring the trustworthiness and safety of deployed models. The presentation will begin with an overview of adversarial attacks on machine learning models and defense strategies for countering the attacks. Common attacks employ adversarially manipulated input instances that are designed to appear perceptually similar to regular data, but are often misclassified by machine learning models. The presentation will next highlight research works from our lab focusing on defense strategies against adversarial examples in medical imaging. Specifically, we will discuss our work on Multi-Instance and Multi-Adversary Robust Self-Training, which aims to improve the robustness of deep learning models for breast ultrasound classification to adversarial images.

3:30 p.m., September 26, 2024

TLC 029

Boyu Zhang, University of Idaho, Dep't of Computer Science

Domain-Specific Vision-Language Model for Automated Skin Cancer Report Generation

Both academia and the medical community have invested significant efforts into developing AI-based automatic skin cancer diagnosis systems using dermoscopy. Artificial intelligence models have achieved promising diagnostic results on large datasets, matching or surpassing human physicians. However, due to a lack of transparency and interpretability in these models' diagnoses, intelligent automatic skin cancer diagnosis has not been widely applied in practice.

This study proposes a multimodal AI model that uses vision-language models to perform comprehensive diagnoses based on input images and meta-information and automatically generates readable diagnostic reports similar to human doctors, including the descriptions of image features corresponding with the commonly used 7-point diagnostic criteria. Experimental results demonstrate that our model can accurately diagnose skin cancer and generate precise, highly readable diagnostic reports. This achievement significantly enhances the interpretability, transparency, and usability of AI diagnostic models. It holds important significance for promoting more widespread early diagnosis of skin cancer and for applications built upon it, such as diagnostic assistants and virtual patients.

3:30 p.m., October 10, 2024

TLC 029

Juan E. Sereno, Postdoctoral Fellow, University of Idaho, Dep't of Mathematics and Statistical Science

Understanding SIR-type epidemic behaviors through set-based dynamic analysis

Abstract: Mathematical models are critical to understand the spread of pathogens in a population and evaluate the effectiveness of non-pharmaceutical interventions. One of the earliest ways to deal with an epidemic is the implementation of so-called social distancing measures, such as: banning of gatherings, university/school closures, stay-at-home measures, mask-wearing, among others.

In this talk we perform a set-based dynamic analysis of SIR-type systems to discuss about realistic implementation of finite-time interval social distancing measures that help us to minimize the Epidemic Final Size, keep the Infected Peak Prevalence controlled at any time, while minimizing the intervention's side effects.

NOTE: Time changed from standard meeting time due to time zone difference with the speaker, who will join us via Zoom from Singapore.

Doudou Zhou, National University of Singapore, Dep't of Statistics and Data Science

Representation Learning for Integrative Analysis of Multi-institutional EHR Data

Abstract: The widespread adoption of electronic health records (EHR) presents unprecedented opportunities for advancing biomedical research and improving patient care. However, integrating data across diverse healthcare systems is challenging due to differences in EHR coding systems, patient demographics, and care models. To overcome these challenges, this talk introduces novel methods for co-training multi-source feature embeddings using (1) block-wise overlapping noisy matrix completion and (2) graph neural networks. These methods address privacy concerns by relying on summary-level EHR data, enabling secure collaboration among institutions. The resulting harmonized embeddings support a range of clinical applications, including cross-institutional code mapping, feature selection, knowledge graph construction, and patient profiling, demonstrating their potential to enhance precision in healthcare decision-making.

Lin Zhang, Simon Fraser University, Department of Statistics and Actuarial Science

Efficient Representation Learning in Large-scale, Heterogeneous Single-cell Omics

Abstract: Single-cell omics data play a pivotal role in identifying cell-to-cell heterogeneity, understanding cell differentiation, unveiling cell population structures, and ultimately deciphering disease pathogenesis.  Due to inherent high-dimensionality, sparsity, noise, and high correlation of single-cell data, machine learning (ML) models, known for its assumption-free flexibility, scalability, and predictive power, have surged in analyzing single-cell data to address these challenges.  In this talk, I will present some of our recent work on ML-based approaches to accurately and efficiently encode single-cell gene expressions and chromatin accessibility.  Our proposed method OCAT, One Cell At A Time, is a ML-based method that sparsely encodes single-cell gene expressions to integrate data from heterogeneous sources without highly variable gene selection or explicit batch effect correction (Wang et al., Genome Biology, 2022).  We have demonstrated that OCAT efficiently integrates multiple heterogeneous scRNA-seq datasets and achieves the state-of-the-art performance in cell type clustering, especially in challenging scenarios of non-overlapping cell types.  In addition, OCAT can efficaciously facilitate a variety of downstream analyses, such as differential gene analysis, trajectory inference, pseudo time inference and cell type inference.  OCAT has proven its efficiency and accuracy in characterizing the transcriptomic difference between healthy and diseased kidney samples (McEvoy et al., Nature Communications, 2022).  We have further developed OCAT2 that maps multiple complementary single-cell omics to the same domain through multi-modal diffusion mapping.  We have demonstrated its accuracy and high computational efficiency on integrating real multi-omics datasets.

Kun Meng, Brown University, Division of Applied Mathematics

Statistical Analysis of Shapes and Images via Euler Characteristics

Abstract:  In the 21st century, we have seen a growing availability of shape-valued and imaging data, prompting the development of new statistical methods to analyze them.  Importantly, bridging the new methods and existing frameworks is advisable.  In this talk, I will introduce several statistical inference methods for shapes and images based on Euler characteristics.  These methods have applications in many fields, such as geometric morphometrics and radiomics.  From a statistical perspective, these methods are naturally connected to functional data analysis and tensor regression.  From a mathematical viewpoint, they are grounded in solid foundations, bridging various branches of mathematics: algebraic and tame topology, Euler calculus, functional analysis, and probability theory.  I will also briefly discuss some of my ongoing and future research directions.

Alexander Panchenko, Washington State University, Department of Mathematics and Statistics

Energy-efficient flocking with non-linear navigational feedback

Abstract: Modeling collective motion in multi-agent systems has gained significant attention. Of particular interest are sufficient conditions for flocking dynamics. We consider a class of dynamical systems that are not necessarily dissipative and lack an obvious Lyapunov function. We address this difficulty by proposing a method to prove the existence of an attractor without relying on LaSalle's principle. Other contributions are as follows. We prove that, under mild conditions, agents' velocities approach the center of mass velocity exponentially, with the distance between the center of mass and the virtual leader being bounded. In the dissipative case, we show existence of a broad class of nonlinear control forces for which the attractor does not contain periodic trajectories, which cannot be rules out by LaSalle's principle. Finally, we conduct a computational investigation of the problem of reducing propulsion energy consumption by selecting appropriate navigational feedback forces.

3:30 p.m., February 22, 2024

TLC 030

Esteban Hernandez-Vargas (University of Idaho)

Redefining the well-posed problems for Calibrating Digital Twins

Abstract:  Jacques Hadamard defined three properties that a well-posed problem should have in order to be interesting for mathematical analysis. While the most important problems in machine learning and in science are inverse problems, these are often not well-posed.

In this talk we will discuss the inverse problem, identifiability, and how we will need to refine math concepts to abstract an Immune Digital Twin, which is a reasonably accurate software replica of the immune system that will revolutionize health space and biology. To reach this dream in the future, we need to start now redefining the inverse problem and engineering computational algorithms.

Tinashe B. Gashirai (U of I, Institute for Modeling Collaboration and Innovation)

Caputo Fractional Derivative and Computational Fractional Dynamics

Abstract: This colloquium seeks to discuss basic concepts related to fractional calculus (FC) in the sense of the Caputo derivative relevant to the application FC and numerical methods for fractional calculus (NMFC) in modeling science and engineering problems that follow non-Markovian processes. Fractional derivatives were introduced over three hundred years ago and the interest in the application of these non-integer order derivatives has significantly grown over the past few decades. These derivatives have an excellent ability of describing (capturing/modeling) complex interactions that possess memory and hereditary traits, among other things. In this presentation, the Caputo fractional derivative (CFD) is preferred over other definitions of non-integer order derivative like the Riemann-Liouville, Riesz and the Grunwald-Letnikov. This is owing to the fact that the interpretation of the initial and boundary conditions of an initial/boundary value problem in the Caputo approach is similar to the case of classical (integer) order approach. This probably explains why the CFD is often utilized in practical applications.

3:30 p.m., March 28, 2024

TLC 030

Michael DePasquale (New Mexico State University)

Connecting the algebra and geometry of line arrangements via rigidity theory

Abstract: A hyperplane arrangement is a union of codimension one linear spaces. These simple objects provide fertile ground for interactions between combinatorics, algebra, algebraic geometry, topology, and group actions. The combinatorics of an arrangement is encoded by the pattern of intersections among the hyperplanes, called its intersection lattice. On the other hand, a key algebraic object is the module of vector fields tangent to the arrangement, called the module of logarithmic derivations, which was introduced by Saito in 1980 to study the singularities of hypersurfaces. An enduring mystery in the theory of hyperplane arrangements is which algebraic properties of the module of logarithmic derivations can be determined from the intersection lattice, and which properties depend fundamentally on the geometry (i.e. the exact hyperplanes). At the center of this mystery is Terao’s conjecture, which proposes that the algebraic property of freeness can be determined only from the intersection lattice. In this talk we explain how rigidity of planar frameworks (dating back to Maxwell) plays a key role in connecting the geometry and algebra of line arrangements in the projective plane. This is joint work with Jessica Sidman and Will Traves.

3:30 p.m., April 4, 2024

TLC 030

Oleksandr Dykhovychnyi (Washington State University)

Energy-efficient flocking with nonlinear navigational feedback

Abstract: Modeling collective motion in multi-agent systems has gained much attention in recent years. In particular, of interest are the conditions under which flocking dynamics emerges. We present a generalization of the multi-agent model of Olfati-Saber with nonlinear navigational feedback forces. As opposed to the original model, our model is, in general, not dissipative. This makes obtaining sufficient conditions for flocking challenging due to the absence of an obvious choice of a Lyapunov function. By means of an alternative argument, we show that our model possesses a global attractor when the navigational feedback forces are bounded perturbations of the linear ones. We further demonstrate that, under mild conditions, the dynamics of the group converges to a complete velocity consensus at an exponential rate. We present a case study of the energy efficiency of our model, illustrating how nonlinear navigational feedback forces, possessing flexibility that linear forces lack, can be used to reduce on-board energy consumption.

3:30 p.m., April 11, 2024

Zoom only

Xinyi Li (Clemson University)

Nonparametric Regression for 3D point Cloud Learning

Abstract: In recent years, there has been an exponentially increased amount of point clouds collected with irregular shapes in various areas. Motivated by the importance of solid modeling for point clouds, we develop a novel and efficient smoothing tool based on multivariate splines over the triangulation to extract the underlying signal and build up a 3D solid model from the point cloud. The proposed method can denoise or deblur the point cloud effectively, provide a multi-resolution reconstruction of the actual signal, and handle sparse and irregularly distributed point clouds to recover the underlying trajectory. In addition, our method provides a natural way of numerosity data reduction. We establish the theoretical guarantees of the proposed method, including the convergence rate and asymptotic normality of the estimator, and show that the convergence rate achieves optimal nonparametric convergence. We also introduce a bootstrap method to quantify the uncertainty of the estimators. Through extensive simulation studies and a real data example, we demonstrate the superiority of the proposed method over traditional smoothing methods in terms of estimation accuracy and efficiency of data reduction.

3:30 p.m., April 18, 2024

TLC 030

Silvia Jimenez Bolanos (Colgate University)

Two recent homogenization results for dielectric elastomer composites

Abstract:  First, we will discuss the periodic homogenization for a weakly coupled electroelastic system of a nonlinear electrostatic equation with an elastic equation enriched with electrostriction. Such coupling is employed to describe dielectric elastomers or deformable (elastic) dielectrics. We will show that the effective response of the system consists of a homogeneous dielectric elastomer described by a nonlinear weakly coupled system of PDEs whose coefficients depend on the coefficients of the original heterogeneous material, the geometry of the composite and the periodicity of the original microstructure. The approach developed here for this nonlinear problem allows obtaining an explicit corrector result for the homogenization of monotone operators with minimal regularity assumptions.

Next, we will discuss the homogenization of high-contrast dielectric elastomer composites. The considered heterogeneous material consisting of an ambient material with inserted particles is described by a weakly coupled system of an electrostatic equation with an elastic equation enriched with electrostriction. It is assumed that particles gradually become rigid as the fine-scale parameter approaches zero. We will see that the effective response of this system entails a homogeneous dielectric elastomer, described by a weakly coupled system of PDEs. The coefficients of the homogenized equations are dependent on various factors, including the composite's geometry, the original microstructure's periodicity, and the coefficients characterizing the initial heterogeneous material. Particularly, these coefficients are significantly influenced by the high-contrast nature of the fine-scale problem's coefficients. Consequently, as anticipated, the high-contrast coefficients of the original yield non-local effects in the homogenized response.

3:30 p.m., April 25, 2024

TLC 030

Jordan Broussard (Whitworth University)

Template Arrays and Two-Dimensional Recurrence Relations

Abstract: In a two-dimensional recurrence relation, there is an underlying structure composed of the two-dimensional sequences (arrays) in which the set of indices is extended from an ordered pair where each entry comes from the set of natural numbers to an ordered pair where each entry comes from the set of integers. The recurrences we will look at have coefficients that come from a field. In this talk, we will look at a set of initial conditions sufficient to build a uniquely-determined array from a given recurrence and initial conditions, as well as look at how to construct a Schauder basis using elementary arrays for the set of arrays.

3:30 p.m., May 2, 2024

Room TBA

3:30 p.m., September 14, 2023

TLC 029

David Andrew Smith (Yale-NUS College - The National University of Singapore)

Fokas Diagonalization

Abstract: We describe a new form of diagonalization for linear two point constant coefficient differential operators with arbitrary linear boundary conditions. Although the diagonalization is in a weaker sense than that usually employed to solve initial boundary value problems (IBVP), we show that it is sufficient to solve IBVP whose spatial parts are described by such operators. We argue that the method described may be viewed as a reimplementation of the Fokas transform method for linear evolution equations on the finite interval. The results are extended to multipoint and interface operators, including operators defined on networks of finite intervals, in which the coefficients of the differential operator may vary between subintervals, and arbitrary interface and boundary conditions may be imposed; differential operators with piecewise constant coefficients are thus included.

3:30 p.m., September 21, 2023

TLC 029

Tuan Phan (University of Idaho, Institute for Interdisciplinary Data Science)

Mathematical modeling of some biological and medical problems in cancer research and physiology

Abstract: In this presentation, I will discuss two mathematical models utilizing stochastic differential equations (SDEs) and/or ordinary differential equations (ODEs). In the first one, we studied human papillomavirus (HPV) infection by proposing a 5-dimensional Ito's SDE system. The theoretical and numerical analyses of the system were conducted to reveal insights intot he progression from HPV infection to cervical cancer. The second one involves cooperative activation of force in human myocardium in which a 3-dimensional ODE system was proposed to capture the relationship between the rate of force redevelopment and relative force in mechanical data from porcine and murine myocardium. The fitting results of the model led to deep understanding of cooperative mechanisms that underlie differences in myocardial contractile dynamics between large and small mammals.

September 28, 2023

TLC 029

Rodolfo Blanco Rodriguez (University of Idaho)

Unraveling multiscale stochastic disease transmission from within-host dynamics to between-host spread

Abstract: In this research, we employ a multiscale modeling approach to unravel the intricate dynamics of infectious disease transmission. Our study spans various scales, from within-host viral interactions to population-level disease dissemination: By employing mathematical models, we explore the replication of viral particles, their impact on infected cells, and the vital role of immune responses, particularly T cells. Moving up the scale, we investigate the transmission of viruses, such as influenza, through different respiratory organs. Zooming out, we analyze disease transmission between individuals in contact networks. Utilizing a stochastic transmission model, we determine the likelihood of infection based on viral levels and immune responses within the contagious host. The timing of host encounters emerges as a critical factor in disease spread.

October 5, 2023

TLC 029

Chencheng Cai (Washington State University)

KoPA: Automated Kronecker Product Approximation

Abstract: We propose to approximate a given matrix by the sum of a few Kronecker products of matrices, which we refer to as the Kronecker product approximation (KoPA). Comparing with the low-rank matrix approximation, KoPA also offers a greater flexibility, since it allows the user to choose the configuration, which are the dimensions of the two smaller matrices forming the Kronecker product. On the other hand, the configuration to be used is usually unknown, and needs to be determined from the data in order to achieve the optimal balance between accuracy and parsimony. We propose to use extended information criteria to select the configuration. Under the paradigm of high dimensional analysis, we show that the proposed procedure is able to select the true configuration with probability tending to one, under suitable conditions on the signal-to-noise ratio. We demonstrate the superiority of KoPA over the low rank approximations through numerical studies, and several benchmark image examples.

October 26, 2023

Via Zoom Only

Michael Allen (Louisiana State University)

Counting number fields generated by plane curves

Abstract: Every irreducible polynomial f(x) with integer coefficients corresponds uniquely to a field extension of the rational numbers which consists Q, a root alpha of f, and all combinations thereof under the standard arithmetic operations.  For example, f(x) = x^2-2 produces the field  Q(sqrt(2)) = {a + b*sqrt(2) : a, b in Q}.  If f is a polynomial in two or more variables, we can produce infinitely many such fields corresponding to solutions to f(x,y)=0.  For f(x,y) = y^2-x^3-x-1, we have solutions (1, sqrt(3)), (2, sqrt(11)), (3, sqrt(31)) and so on, and so the curve defined by f(x,y)=0 “generates” the fields Q(sqrt(3)),  Q(sqrt(11)), and Q(sqrt(31)).

Recently, Mazur and Rubin suggested using this algebraic information as a means to study the geometric properties of a curve.  One can easily ask the reverse question: ``If we know something about a curve C, what can we say about the fields that it generates?"  We approach this question through the lens of arithmetic statistics by counting the number of such fields with bounded size -- under some appropriate notion of size -- for an arbitrary fixed plane curve C.  This is joint work with Renee Bell, Robert Lemke Oliver, Allechar Serrano Lopez, and Tian An Wong.

November 2, 2023

TLC 029

Daryl Robert Deford (Washington State University)

Political Geometries

Abstract: The problem of constructing "fair" political districts and the related problem of detecting intentional gerrymandering has received a significant amount of attention in recent years. Attempting to analyze these issues from a mathematical perspective leads to a wide variety of interesting research problems in geometry, graph theory, and probability. In this talk, I will discuss recent work centered around Markov chain sampling of districting plans that has motivated theoretical questions in these fields, including designing proposal distributions, evaluating the computational complexity of sampling, and measuring the geometric and partisan properties of districts. This work has also helped inform legislative reform efforts and appeared in court challenges, including cases in the Supreme Court this year, and I will discuss what it is like to translate math research to these applied settings and some of the related data,  computational, and communication challenges.

November 16, 2023

TLC 029

Hazem Aboutaleb (University of Idaho)

Causality Verification and Enforcement for Microelectric Package Macromodels

Abstract: The design and analysis phase of passive structures of high-speed microelectronic systems require suitable macromodels that capture the relevant electromagnetic properties that affect the signal and power quality. These models are constructed from either direct measurements, or electromagnetic simulations using macromodeling techniques such as Vector Fitting.

The raw data that are used for extraction of such models have the form of discrete port frequency responses and they may be contaminated by errors due to noise, inadequate calibration techniques in case of direct measurements or approximation and discretization errors in case of numerical simulations. Besides, these data are typically available over a finite frequency range as discrete sets with a limited number of samples. All this may affect the performance of the macromodeling algorithm. Often the underlying cause of such behavior is the lack of causality in given data.

We study the periodic polynomial and spectral continuation methods to enforce causality. Both methods are based on Kramers-Kronig relations, also called dispersion relations. The two methods are successfully tested on several analytic and simulated examples that represent interconnect macromodeling systems to show excellent performance of the proposed techniques.

3:30 p.m., February 23, 2023

TLC 029

Alex Woo (University of Idaho)

The Stanley-Stembridge conjecture and cohomology of Hessenberg varieties

Abstract: The chromatic symmetric function of a graph encodes all of its possible colorings. Richard Stanley and John Stembridge conjectured almost 30 years ago that the chromatic symmetric functions of certain graphs can be written as a positive sum of products of elementary symmetric functions. Each symmetric function can be associated to a formal linear combination of representations of the symmetric group. John Shareshian and Michelle Wachs conjectured that these chromatic symmetric functions are associated to a representation of the symmetric group on the cohomology ring of geometric objects known as Hessenberg varieties; this conjecture was proven by Patrick Brosnan and Timothy Chow and independently by Mathieu Guay-Paquet. By work of Julianna Tymoczko, this cohomology ring can be represented as a vector space on sequences of polynomials satisfying certain relations; the Stanley-Stembridge conjecture, still unsolved, reduces to a statement that this vector space has a basis that is permuted when the sequences of polynomials are themselves permuted in a certain way.

My aim in this talk will be to make the previous paragraph at least somewhat comprehensible. If I talk about any new work, it will be joint work with Erik Insko (Florida Gulf Coast University) and Martha Precup (Washington University in St. Louis).

3:30 p.m., March 30, 2023

TLC 029

Xiongzhi Chen (Washington State University)

Geometry, Topology, Statistics and Learning

Abstract: Doubtlessly, we are now in the “Data Era”. However, unconventional data types, such as shapes, social networks, phylogenetic trees and persistence diagrams, have brought new challenges that learning theory rooted in Euclidean spaces is inadequate to address. Further, the recent rise and success of machine learning techniques have pushed learning into the realm of “Big Models”. These models have complexities and deal with sample sizes that are respectively orders more complicated and larger than those of “high-dimensional models”. This prompts us to rethink about the foundations of statistical learning and its implementation in practice. In this talk, I will give an overview on learning, discuss inference and predictive modelling in general, and explain the role of geometry and topology in statistical learning.

3:30 p.m., April 27, 2023

TLC 029

Abhishek Kaul (Washington State University)

An Efficient Two Step Algorithm for High Dimensional Change Point Regression Models Without Grid Search

Abstract: We propose a two step algorithm based on L1/L0 regularization for the detection and estimation of parameters of a high dimensional change point regression model and provide the corresponding rates of convergence for the change point as well as the regression parameter estimates. Importantly, the computational cost of our estimator is only 2·Lasso(n, p), where Lasso(n, p) represents the computational burden of one Lasso optimization in a model of size (n, p). In comparison, existing grid search based approaches to this problem require a computational cost of at least n · Lasso(n, p) optimizations. Additionally, the proposed method is shown to be able to consistently detect the case of ‘no change’, i.e., where no finite change point exists in the model.  We then characterize the corresponding effects on the rates of convergence of the change point and regression estimates.  Simulations are performed to empirically evaluate performance of the proposed estimators. The methodology is applied to community level socio-economic data of the U.S., collected from the 1990 U.S. census and other sources.

3:30 p.m.,  May 4, 2023

TLC 029

Alejandro Anderson (University of Idaho Postdoctoral Fellow)

Set-control theory: model predictive control and controllable state-space topologies

Abstract: Model Predictive Control (MPC) is a form of control that solves on-line, at each sampling instant, a finite horizon optimal control problem, and only the first input of the optimal control sequence is applied to the real system. The MPC technique has the ability to cope with hard constraints on controls and states and it handles large multivariable systems. Lyapunov theory provides a theoretical framework to prove asymptotic stability of a system controlled by an MPC by employing general concepts of permanence regions of dynamical systems. The permanence regions play an important role in the controllability and stability analysis since such regions (i.e. equilibrium sets, invariant sets, limit cycles, etc.) are the only ones that can be formally stabilized by a controller. In this lecture the importance of general permanence regions on the formulation of stable control strategies such as MPC will be discussed and a methodology for computing these type of regions in the state-space will be presented.