En géométrie algébrique, les invariants de Gromov—Witten sont des invariants énumératifs qui comptent le nombre de courbes complexes dans une variété projective lisse qui vérifient des conditions d’incidence. En 2001, A. Givental et Y.P. Lee ont défini de nouveaux invariants, dits de Gromov—Witten K-théoriques, en remplaçant les définitions cohomologiques dans la construction des invariants de Gromov—Witten par leurs analogues K-théoriques. Une question essentielle est de comprendre comment sont reliées ces deux théories. En 2013, Iritani- Givental-Milanov-Tonita démontrent que les invariants K-théoriques peuvent être encodés dans une fonction qui vérifie des équations aux q-différences. En général, ces équations fonctionnelles vérifient une propriété appelée “confluence”, selon laquelle on peut dégénérer ces équations pour obtenir une équationdifférentielle. Dans cette thèse, on propose de comparer les deux théories de Gromov— Witten à l’aide de la confluence des équations aux q-différences. On montre que, dans le cas des espaces projectifs complexes, que ce principe s’adapte et que les invariants Kthéoriques peuvent être dégénérés pour obtenir leurs analogues cohomologiques. Plus précisément, on montre que la confluence de la petite fonction J de Givental K-théorique permet de retrouver son analogue cohomologique après une transformation par le caractère de Chern. / In algebraic geometry, Gromov— Witten invariants are enumerative invariants that count the number of complex curves in a smooth projective variety satisfying some incidence conditions. In 2001, A. Givental and Y.P. Lee defined new invariants, called Ktheoretical Gromov—Witten invariants. These invariants are obtained by replacing cohomological objects used in the definition of the usual Gromov—Witten invariants by their Ktheoretical analogues. Then, an essential question is to understand how these two theories are related. In 2013, Iritani-Givental- Milanov-Tonita show that K-theoretical Gromov—Witten invariants can be embedded in a function which satisfies a q-difference equation. In general, these functional equations verify a property called “confluence”, which guarantees that we can degenerate these equations to obtain a differential equation. In this thesis, we propose to compare our two Gromov—Witten theories through the confluence of q-difference equations. We show that, in the case of complex projective spaces, this property can be adapted to degenerate Ktheoretical invariants into their cohomological analogues. More precisely, we show that theconfluence of Givental’s small K-theoretical Jfunction produces its cohomological analogue after applying the Chern character.

In this thesis the Lyapunov exponents of random dynamical systems are presented and investigated. The main results are: 1. In the space of all unbounded linear cocycles satisfying a certain integrability condition, we construct an open set of linear cocycles have simple Lyapunov spectrum and no exponential separation. Thus, unlike the bounded case, the exponential separation property is nongeneric in the space of unbounded cocycles. 2. The multiplicative ergodic theorem is established for random difference equations as well as random differential equations with random delay. 3. We provide a computational method for computing an invariant measure for infinite iterated functions systems as well as the Lyapunov exponents of products of random matrices. / In den vorliegenden Arbeit werden Lyapunov-Exponented für zufällige dynamische Systeme untersucht. Die Hauptresultate sind: 1. Im Raum aller unbeschränkten linearen Kozyklen, die eine gewisse Integrabilitätsbedingung erfüllen, konstruieren wir eine offene Menge linearer Kyzyklen, die einfaches Lyapunov-Spektrum besitzen und nicht exponentiell separiert sind. Im Gegensatz zum beschränkten Fall ist die Eingenschaft der exponentiellen Separiertheit nicht generisch in Raum der unbeschränkten Kozyklen. 2. Sowohl für zufällige Differenzengleichungen, als auch für zufällige Differentialgleichungen, mit zufälligem Delay wird ein multiplikatives Ergodentheorem bewiesen. 3.Eine algorithmisch implementierbare Methode wird entwickelt zur Berechnung von invarianten Maßen für unendliche iterierte Funktionensysteme und zur Berechnung von Lyapunov-Exponenten für Produkte von zufälligen Matrizen.

A new method for the determination of finely-gridded reservoir simulation pressures has been developed. It is estimated to be as much as hundreds to thousands of times faster than other methods for very large reservoir simulation grids. The method extends the work of Weber et al. Weber demonstrated accuracies for the pressure solution normally requiring millions of cells using traditional finite-difference equations with only hundreds of cells. This was accomplished through the use of finite-difference equations that incorporate the physics of the flow. Although these coarse-grid solutions achieve accuracies normally requiring orders of magnitude more resolution, their coarse resolution does not resolve local pressure variations resulting from fine-grid permeability variations. Many oil reservoir simulation models require fine grids to adequately represent the reservoir properties. Weber's coarse grids are of little value. This study takes advantage of the accurate coarse-grid solutions of Weber, by nesting them in the requisite fine grids to achieve much faster solutions of the large systems. Application of the nested-grid method involved calculating an accurate solution on a coarse grid, nesting the coarse-grid solution as fixed points into a finer grid and solving. Best results were obtained when an optimal number of coarse-grid pressure points were nested into the fine grid and when an optimal number of nested-grid systems were used.

[ES] Actualmente existe una clara concienciación de la población por la sostenibilidad, el cuidado del medio ambiente y el bienestar animal. Pero, además los consumidores exigen alimentos seguros lo que implica a toda la cadena productiva empezando por la producción primaria. Un adecuado control de las enfermedades transmisibles a este nivel es uno de los pilares fundamentales de la seguridad alimentaria junto con el control en el momento del sacrificio, procesado y distribución. En esta tesis se plantea la utilización de herramientas matemáticas que permitan optimizar el uso de las medidas de bioseguridad, de implantación general en granjas de aves, como son la vacunación, la limpieza y desinfección, y la detección y eliminación de animales infectados. Esto con la fi nalidad de lograr una producción libre de infección y por lo tanto evitar el sacrificio temprano de los animales. De esta forma, se puede contribuir a la sostenibilidad de las granjas. Además, de garantizar la inocuidad de los alimentos a nivel de la producción primaria. Así se ha estudiado el comportamiento de un modelo matemático estructurado que incorpora la contaminación del medio ambiente como un modo indirecto de transmisión de la enfermedad, centrándose en el análisis de un brote de Salmonella en una granja de pollos. Las variables consideradas han sido: individuos susceptibles e infectados y la cantidad de bacterias acumuladas en el recinto (sistema (SIB), y se considera la reposición de los individuos muertos de forma que el tamaño de la población es el mismo en cualquier etapa. El sistema se considera dinámico y no lineal, en tiempo discreto y por ello su modelización se basa en ecuaciones en diferencias. Se ha analizado el comportamiento del sistema alrededor de los puntos de equilibrio a) libre de enfermedad y b) endémico. Tras el análisis del proceso se ha obtenido el número reproductivo básico R0. Este número indica el comportamiento de la enfermedad, ya que si R0 es menor que la unidad, la enfermedad tiende a desaparecer pero en caso contrario la enfermedad permanece endémica o puede llegar a crecer. El resultado obtenido del modelo indica que R0 es menor que uno, si y sólo si, la población se mantiene por debajo de cierto valor umbral, lo que permite tener la enfermedad controlada hacia su desaparición. También, se han estudiado tres modelos para conseguir redirigir la evolución de la enfermedad hacia su desaparición considerando las siguientes medidas: a) vacunación, b) limpieza y desinfección periódica y c) análisis y eliminación periódica de individuos infectados. Los objetivos a alcanzar con el modelo propuesto fueron que la vacunación redujese la incidencia de la enfermedad entre los sujetos susceptibles y determinar su impacto sobre la incidencia. Respecto a la desinfección del recinto y la eliminación de infectados, el objetivo ha sido construir, en cada caso, un nuevo sistema dinámico con coefi cientes periódicos que representase matemáticamente la estrategia de acción periódica elegida. La fi nalidad ha sido optimizar el número de etapas que se puede estar sin actuar sobre el proceso y manteniéndose este estable, es decir con el número reproductivo básico menor que la unidad. Y, por último, se han comparado ambas estrategias, en base a sus periodos máximos. Los resultados obtenidos indican que, respecto a la efectividad de la vacunación, el nuevo número reproductivo básico es función de la tasa de vacunación y de la tasa de efectividad de la vacuna. Si el proceso transcurre con un número reproductivo básico muy alto se requiere vacunar a un mayor número de individuos. Además, cuanto más efectiva sea la vacuna la tasa de vacunación se puede reducir. Para el modelo de impacto la vacunación, se ha indicado que la tasa de vacunación en los programas de vacunación se reduce si el impacto de ésta es positivo reduciendo la tasa de contagios entre los vacunados respecto a la de los susceptibles. / [CA] Actualment existeix una clara conscienciacio de la poblacio per la sostenibilitat, la cura del medi ambient i el benestar animal. Pero, a mes els consumidors exigeixen aliments segurs el que implica a tota la cadena productiva comencant per la produccio primaria. Un adequat control de les malalties transmissibles a aquest nivell es un dels pilars fonamentals de la seguretat alimentaria juntament amb el control en el moment del sacri ci, processament i distribucio. En aquesta tesi es planteja la utilitzacio d'eines matematiques que permeten optimitzar l'us de les mesures de bioseguretat, d'implantacio general en granges d'ocells, com son la vacunacio, la neteja i desinfeccio, i la deteccio i eliminaci o d'animals infectats. Aixo amb la nalitat d'aconseguir una produccio lliure d'infeccio i per tant evitar el sacri ci primerenc dels animals. D'aquesta manera, es pot contribuir a la sostenibilitat de les granges. A mes, de garantir la innocu tat dels aliments a nivell de la produccio primaria. Aix, s'ha estudiat el comportament d'un model matematic estructurat que incorpora la contaminacio del medi ambient com una manera indirecta de transmissio de la malaltia, centrant-se en l'analisi d'un brot de Salmonella en una granja de pollastres. Les variables considerades han sigut: individus susceptibles i infectats i la quantitat de bacteris acumulats en el recinte (sistema SIB), i, a mes, es considera reposicio dels individus morts de manera que la grandaria de la poblacio es el mateix en qualsevol etapa. El sistema es considera dinamic i no lineal, en temps discret i per aixo la seua modelitzacio es basa en equacions en diferencies. S'ha analitzat el comportament del sistema al voltant dels punts d'equilibri a) lliure de malaltia i b) endemic. Despres de l'analisi del proces s'ha obtingut el numero reproductiu basic R0. Aquest numero indica el comportament de la malaltia, ja ix que si R0. es menor que la unitat, la malaltia tendeix a desapareixer pero en cas contrari la malaltia roman endemica o pot arribar a creixer. El resultat obtingut del model indica que R0. es menor que un, si i nomes si, la poblacio es mante per davall d'un cert valor llindar, la qual cosa permet tindre la malaltia controlada cap a la seua desaparicio. Tambe, s'han estudiat tres models per a aconseguir redirigir l'evolucio de la malaltia cap a la seua desaparicio considerant les seg uents mesures: a) vacunacio, b) neteja i desinfeccio periodica i c) analisi i eliminacio periodica d'individus infectats. Els objectius a aconseguir amb el model proposat van ser que la vacunacio redu ra la incidencia de la malaltia entre els subjectes susceptibles i determinar el seu impacte sobre la incidencia. Respecte a la desinfeccio del recinte i l'eliminacio d'infectats, l'objectiu ha sigut construir, en cada cas, un nou sistema dinamic amb coe cients periodics que representara matematicament l'estrategia d'accio periodica triada. La nalitat ha sigut optimitzar el nombre d'etapes que es pot estar sense actuar sobre el proces i mantenint-se aquest estable, es a dir amb el numero reproductiu basic menor que la unitat. I, nalment, s'han comparat totes dues estrategies, sobre la base dels seus perodes maxims. Els resultats obtinguts indiquen que, respecte a l'efectivitat de la vacunacio, el nou numero reproductiu basic es funcio de la taxa de vacunacio i de la taxa d'efectivitat de la vacuna. Si el proces transcorre amb un numero reproductiu basic molt alt es requereix vacunar a un major nombre d'individus. A mes, com mes efectiva siga la vacuna la taxa de vacunacio es pot reduir. Per al model d'impacte la vacunacio, s'ha indicat que la taxa de vacunacio en els programes de vacunacio es redueix si l'impacte d'aquesta es positiu reduint la taxa de contagis entre els vacunats respecte a la dels susceptibles. / [EN] There is currently a clear awareness of the population for sustainability, care for the environment and animal welfare. But in addition, consumers demand safe food, which involves the entire production chain, starting with primary production. Adequate control of communicable diseases at this level is one of the fundamental pillars of food security along with control at the time of slaughter, processing and distribution. This thesis proposes the use of mathematical tools to optimize the use of biosafety measures, general implementation in bird farms, such as vaccination, cleaning and disinfection, and the detection and elimination of infected animals. This in order to achieve an infection free production and therefore avoid early slaughter of animals. In this way, it can contribute to the sustainability of farms. In addition, to ensure food safety at the level of primary production. Thus, the behavior of a structured mathematical model that incorporates environmental pollution as an indirect mode of disease transmission has been studied, focusing on the analysis of a Salmonella outbreak on a farm. chickens. The variables considered were susceptible and infected individuals and the amount of bacteria accumulated in the enclosure (SIB system), and, in addition, replacement of dead individuals is considered so that the size of the population is the same at any stage. The system is considered dynamic and nonlinear, in discrete time and therefore its modeling is based on equations in dierences. The behavior of the system around the equilibrium points a) free of disease and b) endemic has been analyzed. After the analysis of the process the basic reproductive number R0. was obtained. This number indicates the behavior of the disease, as if R0 is less than unity, the disease tends to disappear but otherwise the disease remains xiii endemic or may grow. The result obtained from the model indicates that R0 is less than one, if and only if, the population remains below a certain threshold value, which allows to have the disease controlled towards its disappearance. Also, three models have been studied to redirect the evolution of the disease towards its disappearance considering the following measures: a) vaccination, b) periodic cleaning and disinfection and c) periodic analysis and elimination of infected individuals. The objectives to be achieved with the proposed model were that vaccination would reduce the incidence of the disease among susceptible subjects and determine its impact on the incidence. Regarding the disinfection of the enclosure and the elimination of infected, the aim has been to build, in each case, a new dynamic system with periodic coecients that will mathematically represent the chosen periodic action strategy. The aim has been to optimize the number of stages that can be left without acting on the process and keeping it stable, ie with the basic reproductive number less than the unit. And nally, both strategies have been compared, based on their maximum periods. The results obtained indicate that, with respect to the eectiveness of vaccination, the new basic reproductive number is a function of the vaccination rate and the vaccine eectiveness rate. If the process proceeds with a very high basic reproductive number it is required to vaccinate a larger number of individuals. In addition, the more eective the vaccine, the lower the vaccination rate. For the vaccination impact model, it has been indicated that the vaccination rate in vaccination programs is reduced if the impact of this is positive by reducing the rate of transmission among vaccinated compared to those susceptible. / Poveda Giner, JJ. (2022). Análisis de procesos epidemiológicos mediante modelos matemáticos: aplicación a la seguridad alimentaria [Tesis doctoral]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/182456

Este trabalho é em parte baseado no livro The Stability and Control of Discrete Processes de Joseph P. LaSalle. Nós estudamos equações como x(n+1) = T(x(n)), onde T : \' R POT. m\' \' SETA\' \'R POT. m\' é uma aplicação contínua, com o sistema dinâmico associado \'PI\' (n,x) := \' T POT. n\' (x). Nós fornecemos condições suficientes para a estabilidade de equilíbrios usando o método direto de Liapunov. Também consideramos sistemas discretos da forma x(n+1)=T(n, x(n),\'lâmbda\' ) dependendo de uma parâmetro \' lâmbda\' e apresentamos resultados obtendo estimativas de atratores. Finalmente, nós apresentamos algumas simulações de sistemas acoplados como uma aplicação em sistemas de comunicação / This work is in part based on the book The Stability and Control of Discrete Processes of Joseph P. LaSalle. We studing equations as x(n+1) = T(x(n)), where T : \' R POT.m\' \' ARROW\' \' \' R POT.m\' is continuous transformation, with the associated dynamic system \'PI\' (n,x) := \' T POT.n\' (x). We provide suddicient conditions for stability of equilibria, using Liapunov direct method. We also consider nonautonomous discrete systems of the form x(n + 1) = T(n, x(n), \' lâmbda\') depending on the parameter \'lâmbda\' and present results obtaining uniform estimatives of attractors. We finally we present some simulations on synchronization of coupled systems as an application on communication systems

Este trabalho é em parte baseado no livro The Stability and Control of Discrete Processes de Joseph P. LaSalle. Nós estudamos equações como x(n+1) = T(x(n)), onde T : \' R POT. m\' \' SETA\' \'R POT. m\' é uma aplicação contínua, com o sistema dinâmico associado \'PI\' (n,x) := \' T POT. n\' (x). Nós fornecemos condições suficientes para a estabilidade de equilíbrios usando o método direto de Liapunov. Também consideramos sistemas discretos da forma x(n+1)=T(n, x(n),\'lâmbda\' ) dependendo de uma parâmetro \' lâmbda\' e apresentamos resultados obtendo estimativas de atratores. Finalmente, nós apresentamos algumas simulações de sistemas acoplados como uma aplicação em sistemas de comunicação / This work is in part based on the book The Stability and Control of Discrete Processes of Joseph P. LaSalle. We studing equations as x(n+1) = T(x(n)), where T : \' R POT.m\' \' ARROW\' \' \' R POT.m\' is continuous transformation, with the associated dynamic system \'PI\' (n,x) := \' T POT.n\' (x). We provide suddicient conditions for stability of equilibria, using Liapunov direct method. We also consider nonautonomous discrete systems of the form x(n + 1) = T(n, x(n), \' lâmbda\') depending on the parameter \'lâmbda\' and present results obtaining uniform estimatives of attractors. We finally we present some simulations on synchronization of coupled systems as an application on communication systems

Στην παρούσα διατριβή παρουσιάζεται η μελέτη μιας οικογένειας εξισώσεων διαφορών (ή διακριτών εξισώσεων) χρησιμοποιώντας μεθόδους συμμετριών. Τέτοιες μέθοδοι είναι καλά θεμελιωμένες για την μελέτη και κατασκευή λύσεων διαφορικών εξισώσεων. Στόχος είναι η χρήση συμμετριών για τη σύνδεση διαφορικών και διακριτών εξισώσεων, καθώς και η κατασκευή λύσεων των τελευταίων από συμμετρικές λύσεις των πρώτων. Συγκεκριμένα, μελετάμε διακριτές εξισώσεις που είναι αφινικά γραμμικές, έχουν τις συμμετρίες του τετραγώνου και εμπλέκουν τέσσερεις τιμές μιας άγνωστης συνάρτησης δύο ακέραιων μεταβλητών, οι οποίες σχηματιζούν ένα στοιχειώδες τετράπλευρο στο επίπεδο των ανεξάρτητων μεταβλητών. Η διεξοδική μελέτη αυτής της οικογένειας οδηγεί στην κατασκευή ενός νόμου διατήρησης καθώς και σε συνθήκες γραμμικοποιήσης. Μέλη αυτής της οικογένειας είναι και οι ολοκληρώσιμες εξισώσεις της ταξινόμησης των Adler, Bobenko, Suris (ABS). Η ολοκληρωσιμότητα των εξισώσεων ABS προκύπτει από την πολυδιάστατη συμβατότητά τους. Αυτό σημαίνει ότι μπορούν να επεκταθούν κατάλληλα σε εξισώσεις πολλών ανεξάρτητων μεταβλητών. Η ιδιότητα αυτή μας επιτρέπει να κατασκευάσουμε άμεσα έναν αυτομεταχηματισμό Bäcklund και ένα ζευγάρι Lax χρησιμοποιώντας τις ίδιες τις εξισώσεις, στοιχεία που αποτελούν άλλη μια ένδειξη της ολοκληρωσιμότητάς τους. Η εξάρτηση των εξισώσεων ABS από δύο συνεχείς παραμέτρους μας επιτρέπει να μελετήσουμε επιπλέον και τις επεκταμένες συμμετρίες τους, δηλαδή τις συμμετρίες που δρουν και στις παραμέτρους. Αυτές οι συμμετρίες αποτελούν το βασικό εργαλείο για τη σύνδεσή τους με ολοκληρώσιμα συστήματα διαφορικών εξισώσεων. Την ολοκληρωσιμότητα αυτών των συμβατών διαφορικών συστημάτων την αποδεικνύουμε κατασκευάζοντας έναν αυτομετασχηματισμό Bäcklund και ένα ζευγάρι Lax. Η σύνδεση αυτή μας επιτρέπει να κατασκευάσουμε λύσεις των διακριτών εξισώσεων από λύσεις του συμβατού συστήματος διαφορικών εξισώσεων, οι οποίες συνδέονται με λύσεις των συνεχών εξισώσεων Painlevé. Από την άλλη, παρουσιάζεται η σύνδεση αυτών των συστημάτων διαφορικών εξισώσεων με τις γεννήτριες εξισώσεις. Οι τελευταίες παρουσιάστηκαν αρχικά από τους Nijhoff, Hone, Joshi χρησιμοποιώντας άλλη προσέγγιση. Ωστόσο, η προσέγγιση μέσω συμμετρικών αναγωγών που παρουσιάζουμε εδώ είναι πιο άμεση και οδηγεί στα ίδια συμπεράσματα. Συνοψίζοντας, η παρούσα διατριβή παρουσιάζει μια καινοτομική χρήση των συμμετριών των διακριτών εξισώσεων για την κατασκευή λύσεων, αλλά και την σύνδεσή τους με συστήματα διαφορικών εξισώσεων. / In the present dissertation, we present the study of a family of discrete equations using symmetry-based techniques. Such methods are well established for the study of differential equations. We use the symmetries of discrete equations to establish new connections between discrete and differential equations, as well as to construct new solutions of the former in terms of similarity solutions of the latter. Specifically, we study discrete equations which are affine linear, possess the symmetries of the square and involve four values of an unknown function of two independent discrete variables forming a quadrilateral. The extensive study of this class leads to a conservation law, as well as to linearization conditions. Members of this family are the integrable equations of the Adler, Bobenko, Suris (ABS) classification. The integrability of the ABS equations follows from their multidimensional consistency. The latter implies that, the equation may be extended in a multidimensional lattice. This property allows us to derive directly an auto– Bäcklund transformation and a Lax pair, using the function defining these equations. These are another evidence of the integrability of the ABS equations. The dependence of these equations on two continuous parameters permits us to study their extended symmetries, i.e. symmetries acting on the parameters as well. These symmetries are our main tool in connecting the ABS equations to integrable systems of differential equations. The integrability of the latter is proved by the construction of an auto–Bäcklund transformation and a Lax pair. This connection provides us the means to construct solutions of the discrete equations from solutions of the compatible differential system, which are related to solutions of the continuous Painlevé equations. On the other hand, we present how these systems lead naturally to generating differential equations, which were presented by Nijhoff, Hone and Joshi starting from another point of view. However, our construction through symmetry reductions is more straightforward. Thus, in the present thesis is presented a novel usage of the symmetries of discrete equations in the construction of solutions and the connection between discrete and differential equations.

Dizertační práce se zabývá konstrukcí obecného řešení slabě zpožděných systémů lineárních diskrétních rovnic v ${\mathbb R}^3$ tvaru \begin{equation*} x(k+1)=Ax(k)+Bx(k-m), \end{equation*} kde $m>0$ je kladné celé číslo, $x\colon \bZ_{-m}^{\infty}\to\bR^3$, $\bZ_{-m}^{\infty} := \{-m, -m+1, \dots, \infty\}$, $k\in\bZ_0^{\infty}$, $A=(a_{ij})$ a $B=(b_{ij})$ jsou konstantní $3\times 3$ matice. Charakteristické rovnice těchto systémů jsou identické s charakteristickými rovnicemi systému, který neobsahuje zpožděné členy. Jsou získána kriteria garantující, že daný systém je slabě zpožděný a následně jsou tato kritéria specifikována pro všechny možné případy Jordanova tvaru matice $A$. Systém je vyřešen pomocí metody, která ho transformuje na systém vyšší dimenze, ale bez zpoždění \begin{equation*} y(k+1)=\mathcal{A}y(k), \end{equation*} kde ${\mathrm{dim}}\ y = 3(m+1)$. Pomocí metod lineární algebry je možné najít Jordanovy formy matice $\mathcal{A}$ v závislosti na vlastních číslech matic $A$ and $B$. Tudíž lze nalézt obecné řešení nového systému a v důsledku toho pak odvodit obecné řešení počátečního systému.

Dizertační práce se zabývá slabě zpožděnými lineárními rovinnými systémemy s konstantními koeficienty. Charakteristická rovnice těchto systémů je identická s charakteristickou rovnicí systému, který neobsahuje zpožděné členy. V takovém případě se počáteční dimenze prostoru řešení mění po několika krocích na menší. V jistém smyslu je tato situace analogická podobnému jevu v teorii lineárních diferenciálních systémů s konstantními koeficienty a speciálním zpožděním, kdy původně nekonečně rozměrný prostor řešení (na počátečním intervalu) přejde po několika krocích do konečného prostoru řešení. V práci je pro každý možný případ kombinace kořenů charakteristické rovnice konstruováno obecné řešení daného systému a jsou formulovány výsledky o dimenzi prostoru řešení. Také je zkoumána stabilita řešení.

<p>Deep ‘Analog Artificial Neural Networks’ (AANNs) perform complex classification problems with high accuracy. However, they rely on humongous amount of power to perform the calculations, veiling the accuracy benefits. The biological brain on the other hand is significantly more powerful than such networks and consumes orders of magnitude less power, indicating some conceptual mismatch. Given that the biological neurons are locally connected, communicate using energy efficient trains of spikes, and the behavior is non-deterministic, incorporating these effects in Artificial Neural Networks (ANNs) may drive us few steps towards a more realistic neural networks. </p> <p> </p> <p>Emerging devices can offer a plethora of benefits including power efficiency, faster operation, low area in a vast array of applications. For example, memristors and Magnetic Tunnel Junctions (MTJs) are suitable for high density, non-volatile Random Access Memories when compared with CMOS implementations. In this work, we analyze the possibility of harnessing the characteristics of such emerging devices, to achieve neuro-inspired solutions to intricate problems.</p> <p> </p> <p>We propose how the inherent stochasticity of nano-scale resistive devices can be utilized to realize the functionality of spiking neurons and synapses that can be incorporated in deep stochastic Spiking Neural Networks (SNN) for image classification problems. While ANNs mainly dwell in the aforementioned classification problem solving domain, they can be adapted for a variety of other applications. One such neuro-inspired solution is the Cellular Neural Network (CNN) based Boolean satisfiability solver. Boolean satisfiability (k-SAT) is an NP-complete (k≥3) problem that constitute one of the hardest classes of constraint satisfaction problems. We provide a proof of concept hardware based analog k-SAT solver that is built using MTJs. The inherent physics of MTJs, enhanced by device level modifications, is harnessed here to emulate the intricate dynamics of an analog, CNN based, satisfiability (SAT) solver. </p> <p> </p> <p>Furthermore, in the effort of reaching human level performance in terms of accuracy, increasing the complexity and size of ANNs is crucial. Efficient algorithms for evaluating neural network performance is of significant importance to improve the scalability of networks, in addition to designing hardware accelerators. We propose a scalable approach for evaluating Liquid State Machines: a bio-inspired computing model where the inputs are sparsely connected to a randomly interlinked reservoir (or liquid). It has been shown that biological neurons are more likely to be connected to other neurons in the close proximity, and tend to be disconnected as the neurons are spatially far apart. Inspired by this, we propose a group of locally connected neuron reservoirs, or an ensemble of liquids approach, for LSMs. We analyze how the segmentation of a single large liquid to create an ensemble of multiple smaller liquids affects the latency and accuracy of an LSM. In our analysis, we quantify the ability of the proposed ensemble approach to provide an improved representation of the input using the Separation Property (SP) and Approximation Property (AP). Our results illustrate that the ensemble approach enhances class discrimination (quantified as the ratio between the SP and AP), leading to improved accuracy in speech and image recognition tasks, when compared to a single large liquid. Furthermore, we obtain performance benefits in terms of improved inference time and reduced memory requirements, due to lower number of connections and the freedom to parallelize the liquid evaluation process.</p>