Academic literature on the topic 'Projection operator'

Principal components analysis (PCA) is effective at compressing information in multivariate data sets by computing orthogonal projections that maximize the amount of data variance. Unfortunately, information content in hyper spectral images does not always coincide with such projections. We propose an application of projection pursuit (pp), which seeks to find a set of projections that are "interesting" in the sense that they deviate from the Gaussian distribution assumption. Once these projections are obtained, they can be used for image compression, segmentation, or enhancement for visual analysis. To find these projections, a two –step iterative process is followed where we first search for a projection that maximizes a projection index based on the information divergence of the projections estimated probability distribution from the Gaussian distribution and then reduce the rank by projections the data on to the subspace orthogonal to the previous projection . To calculate each projections, we use a simplified approach to maximizing the projection index, which does not require optimization algorithm. It searches for a solution by obtaining a set of candidate projections from the data and choosing the one with the highest projection index. The effectiveness of the method is demonstrated through simulated examples as well as data from the hyper spectral digital imagery collection experiment and the spatially enhanced broadband and array spectrograph system.

We give a necessary and sufficient condition to determine when an operator in the nest algebra of doubly infinite block upper triangular operators factors through a diagonal projection. An example shows that this condition does not extend to more general nest algebras, but a similar criterion yields a description of the ideals of nest algebras generated by diagonal projections.

In the noncommutative phase space, a new mapping is proposed to express the noncommutative coordinate and momentum operators in terms of the ordinary coordinate and momentum operators under the case of large noncommutativity parameters (μν>1). Using this mapping matrix, the deformed boson operators can be expressed in terms of the ordinary boson operators. Thus, the normal ordering expansion form of vacuum projection operator is obtained. As an application, the completeness relation of the two-mode deformed coherent states is verified by using the vacuum projection operator.

In a recent work, Bringmann and Ono [4] show that Ramanujan's f(q) mock theta function is the holomorphic projection of a harmonic weak Maass form of weight 1/2. In this paper, we extend the work of Ono in [13]. In particular, we study holomorphic projections of certain integer weight harmonic weak Maass forms on SL 2(ℤ) using Hecke operators and the differential theta-operator.

A data-driven analysis method known as dynamic mode decomposition (DMD) approximates the linear Koopman operator on a projected space. In the spirit of Johnson–Lindenstrauss lemma, we will use a random projection to estimate the DMD modes in a reduced dimensional space. In practical applications, snapshots are in a high-dimensional observable space and the DMD operator matrix is massive. Hence, computing DMD with the full spectrum is expensive, so our main computational goal is to estimate the eigenvalue and eigenvectors of the DMD operator in a projected domain. We generalize the current algorithm to estimate a projected DMD operator. We focus on a powerful and simple random projection algorithm that will reduce the computational and storage costs. While, clearly, a random projection simplifies the algorithmic complexity of a detailed optimal projection, as we will show, the results can generally be excellent, nonetheless, and the quality could be understood through a well-developed theory of random projections. We will demonstrate that modes could be calculated for a low cost by the projected data with sufficient dimension.

The computed tomography allows to reconstruct the inner morphological structure of an object without physical destructing. The accuracy of digital image reconstruction directly depends on the measurement conditions of tomographic projections, in particular, on the number of recorded projections. In medicine, to reduce the dose of the patient load there try to reduce the number of measured projections. However, in a few-view computed tomography, when we have a small number of projections, using standard reconstruction algorithms leads to the reconstructed images degradation. The main feature of our approach for few-view tomography is that algebraic reconstruction is being finalized by a neural network with keeping measured projection data because the additive result is in zero space of the forward projection operator. The final reconstruction presents the sum of the additive calculated with the neural network and the algebraic reconstruction. First is an element of zero space of the forward projection operator. The second is an element of orthogonal addition to the zero space. Last is the result of applying the algebraic reconstruction method to a few-angle sinogram. The dependency model between elements of zero space of forward projection operator and algebraic reconstruction is built with neural networks. It demonstrated that realization of the suggested approach allows achieving better reconstruction accuracy and better computation time than state-of-the-art approaches on test data from the Low Dose CT Challenge dataset without increasing reprojection error.

Abstract We introduce concepts of essential numerical range for the linear operator pencil $\lambda \mapsto A-\lambda B$. In contrast to the operator essential numerical range, the pencil essential numerical ranges are, in general, neither convex nor even connected. The new concepts allow us to describe the set of spectral pollution when approximating the operator pencil by projection and truncation methods. Moreover, by transforming the operator eigenvalue problem $Tx=\lambda x$ into the pencil problem $BTx=\lambda Bx$ for suitable choices of $B$, we can obtain nonconvex spectral enclosures for $T$ and, in the study of truncation and projection methods, confine spectral pollution to smaller sets than with hitherto known concepts. We apply the results to various block operator matrices. In particular, Theorem 4.12 presents substantial improvements over previously known results for Dirac operators while Theorem 4.5 excludes spectral pollution for a class of nonselfadjoint Schrödinger operators which has not been possible to treat with existing methods.

AbstractThe distance from an arbitrary rank-one projection to the set of nilpotent operators, in the space of k × k matrices with the usual operator norm, is shown to be sec(π/(k:+2))/2. This gives improved bounds for the distance between the set of all non-zero projections and the set of nilpotents in the space of k × k matrices. Another result of note is that the shortest distance between the set of non-zero projections and the set of nilpotents in the space of k × k matrices is .

Background There is increasing concern and focus in the interventional cardiology community on potential long term health issues related to radiation exposure and heavy wearable protection. Optimized shielding measures may reduce operator dose to levels where lighter radioprotective garments can safely be used, or even omitted. X-ray blankets (XRB) are commercially available but suffer from small size and lack of stability. A larger XRB may reduce operator dose but could hamper vascular access and visualization. The aim of this study is to assess shielding effect of an optimized XRB during cardiac catheterization and estimate the potential reduction in annual operator dose based on DICOM Radiation Dose Structured Report (RDSR) data reflecting everyday clinical practice. Methods Data accumulated from 7681 procedures over three years in our RDSR repository was used to identify projection angles and radiation doses during cardiac catheterization. Using an anthropomorphic phantom and a scatter radiation detector, radiation dose to the operator (mSv) and patient (dose area product—DAP) was measured for each angiographic projection for three different shielding setups. Relative operator dose (mSv/DAP) was calculated and multiplied by DAP per projection to estimate effect on operator dose. Results Adding an optimized XRB to a standard shielding setup comprising a table- and ceiling-mounted shield resulted in a 94.9% reduction in estimated operator dose. The largest shielding effect was observed in left and cranial projections where the ceiling-mounted shield offered less protection. Conclusions An optimized XRB is a simple shielding measure that has the potential to reduce operator dose.

Density functional theory (DFT) is widely used to describe the electronic structure of molecular systems and, thanks to the simplicity of its theoretical framework, it is particularly suitable for the development quantum embedding schemes. In this dissertation a novel embedding scheme, based on the employment of a projection operator, is presented. This method allows one to embed one sub-region of a given molecular system in its environment and treat these regions at different level of theory (e.g. CCSD(T)in- DFT). Thanks to the use of a projection technique that enforces the Pauli principle between subsystems, the complications associated with the appearance of non-additive kinetic energy contributions are overcome. First, I show a general software implementation of the method and the features that allow the analysis of a variety of chemical problems (e.g. organic reactions, transition metal complexes). Next, I apply the method to a wide range of benchmarking examples chosen to assess accuracy and performance. Namely, the SN2 reaction of I-propylchloride with CI- , phenol molecule deprotonation reaction, association of' iron(II) to ethylamine, Diels-Alder cycloaddition, and Stone' Vales rotation reaction are investigated. I show that , for such examples, this framework is able to reproduce the accuracy of highly correlated wave-function (WF) methods with reduced computational cost, by performing WF-in-DFT calculations. In addition, by exploring several simulation conditions, e.g. different functionals, localisation schemes, basis sets, I demonstrate the performance of the method displays a fairly independent behaviour with respect to simulation conditions. Finally, once the robustness of the code has been tested, I extend applications to more realistic chemical systems of technological and experimental interest, i.e. adsorption of cobalt on coronene. A further improvement of the method is also described. I assess a new version of the code that enables further reduction of the computational cost and the possibility of enlarging the size of the systems studied by performing an intelligent truncation of the atomic basis set used in the WF-based calculation.

Human operator models developed using optimal control theory are typically complicated and over-parameterized, even for simple controlled elements. Methods for generating less complicated operator models that preserve the most important characteristics of the full order model are developed so that the essential features of the operator dynamics are easier to determine. A new formulation of the Optimal Control Model (OCM) of the human operator is developed that allows order reduction techniques to be applied in a meaningful way. This formulation preserves the critical neuromotor dynamics and time delay characteristics of the human operator. The Optimal Projection (OP) synthesis technique is applied to a modified version of the OCM. Using OP synthesis allows one to determine operator models that minimize the quadratic performance index of the OCM with a constraint on model order. This technique allows analysts to formulate operator models of fixed order. Operator model reduction methods based on variations of balanced realization techniques are also developed since they reduce the computational complexity associated with OP synthesis yet maintain a reasonable level of accuracy. Computer algorithms are developed that insure that the reduced order models have noise to signal ratios that are consistent with OCM theory. The OP method generates operator models of fixed order that are consistent with OCM theory in all respects, i.e. optimality, neuromotor lag, time delay, and noise to signal ratios are all preserved. The other model reduction techniques preserve these features with the exception of optimality. Each technique is applied to a variety of controlled elements to illustrate how performance and frequency response fidelity degrade when the order of the operator model is reduced.
Ph. D.

Die vorliegende Arbeit ist der Untersuchung von Operatorfolgen gewidmet, die typischerweise bei der Anwendung von Approximationsverfahren auf stetige lineare Operatoren entstehen. Dabei stehen die Stabilität der Folgen sowie das asymptotische Verhalten gewisser Charakteristika wie Normen, Konditionszahlen, Fredholmeigenschaften und Pseudospektren im Mittelpunkt. Das Hauptaugenmerk liegt auf der Entwicklung der Theorie für Operatoren auf Banachräumen. Hierbei bildet ein dafür geeigneter Konvergenzbegriff, die sogenannte P-starke Konvergenz, den Ausgangspunkt, welcher das Studium der gewünschten Eigenschaften in einer erstaunlichen Allgemeinheit gestattet. Die erzielten Resultate kommen, neben einer Reihe weiterer Anwendungen, insbesondere für das Projektionsverfahren für banddominierte Operatoren zum Einsatz.

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CAPES - Coordenação de Aperfeiçoamento de Pessoal de Nível Superior
Neste trabalho, o objetivo é estudar a existência e unicidade da solução num sentido fraco para um problema não linear com valor na fronteira que é derivado de um modelo que decreve o equilibrio de um plasma confinado. Para esta finalidade se formula um problema equivalente e se estabelecem condições para este novo problema. Logo, utilizando a teoria da subdiferencial e fazendo um estudo de autovalor se consegue que este novo problema tenha solução e, além disso, seja única.
In this work, the objective is to study the existence and uniqueness of the solution in a weak sense of a nonlinear boundary value problem which it is derived from a model that describe the equilibrium of a confined plasma. For this purpose, we formulate an equivalent problem and establish conditions for this new problem. Therefore, using the theory of subdiferencial and studing an eigenvalue problem, we obtain that this new problem has a unique solution.

The aim of this thesis is to give a mathematical framework for scattering of electromagnetic waves by rough surfaces. We prove that the Maxwell transmission problem with a weakly Lipschitz interface,in finite energy norms, is well posed in Fredholm sense for real frequencies. Furthermore, we give precise conditions on the material constants ε, μ and σ and the frequency ω when this transmission problem is well posed. To solve the Maxwell transmission problem, we embed Maxwell’s equations in an elliptic Dirac equation. We develop a new boundary integral method to solve the Dirac transmission problem. This method uses a boundary integral operator, the rotation operator, which factorises the double layer potential operator. We prove spectral estimates for this rotation operator in finite energy norms using Hodge decompositions on weakly Lipschitz domains. To ensure that solutions to the Dirac transmission problem indeed solve Maxwell’s equations, we introduce an exterior/interior derivative operator acting in the trace space. By showing that this operator commutes with the two basic reflection operators, we are able to prove that the Maxwell transmission problem is well posed. We also prove well-posedness for a class of oblique Dirac transmission problems with a strongly Lipschitz interface, in the L_2 space on the interface. This is shown by employing the Rellich technique, which gives angular spectral estimates on the rotation operator.

Cette thèse porte sur l'étude de classes d'opérateurs. On étudie principalement deux familles différentes de classes d'opérateurs.- Les premières classes étudiées sont des classes d'opérateurs sur des espaces de Hilbert généralisant les classes $C_{ho}$ de Sz.Nagy et Foias.Pour $(ho_n)_n$ une suite de nombres complexes non-nuls, on définit la classe $C_{(ho_n)}(H)$ comme l'ensemble des opérateurs $T in mathcal{L}(H)$ qui possèdent une $(ho_n)$-dilatation : il existe un espace de Hilbert K et un opérateur unitaire $U in mathcal{L}(K)$ avec $H subset K$ tels que $T^n=ho_n P_H U^n|_H$ pour tout n $geq$ 1 ($P_H in mathcal{L}(K)$ étant la projection orthogonale de K sur H). Ces classes peuvent être associées à une fonction holomorphe $f_{(ho_n)}$ ainsi qu'à une quasi-norme $w_{(ho_n)}$. Nous utilisons les liens entre ces trois objets pour caractériser, décrire, et donner plusieurs propriétés spectrales sur les opérateurs contenues dans ces classes. Nous exhibons de même des relations entre plusieurs classes de cette forme, nous généralisons des résultats connus pour les classes $C_{(ho)}$, et donnons divers exemples et situations offrant des comportements différents du cas $C_{(ho)}$. Nous apportons aussi une nouvelle vision géométrique sur un résultat entre des quasi-normes $w_{ho}$, et nous étendons des calculs de $w_{ho}(T)$ pour des opérateurs T annulés par un polynôme de degré deux.- La deuxième partie principale de cette thèse concerne les classes de L^p-projections. Une L^p-projection sur un espace de Banach X, pour $1leq p leq +infty$, est une projection P qui vérifie $ |f|_X = |(|P(f)|_X, |(I-P)(f)|_X) |_{ell_{p}}$ pour tout f dans X. Cette relation est une version L^p de l'égalité $|f|^2=|Q(f)|^2 + |(I-Q)(f)|^2$, vérifiée pour les projections orthogonales dans les espaces de Hilbert.Nous nous intéressons aux relations entre les L^p-projections sur un espace de Banach X et celles sur un sous-espace F, sur un quotient X/F, ou sur un sous-espace de quotient G/F. Des caractérisations complètes sont apportées pour des espaces de Banach vérifiant quelques propriétés additionnelles, et selon la valeur de p.Nous introduisons aussi la notion de L^p-projection maximale pour X, c'est-à-dire des L^p-projections définies sur un sous-espace G de X qui ne peuvent pas être étendues comme L^p-projections sur un sous-espace plus grand, et étudions leurs propriétés, en particulier dans le cas de la dimension finie.Nous obtenons de même une caractérisation des L^{infty}-projections sur tous les espaces L^{infty}(Omega) via de nouvelles méthodes, en généralisant ainsi les résultats connus à ce sujet
This thesis focuses on the study of classes of operators. Two different families of classes of operators are mainly studied.- The first classes we study are classes of operators on Hilbert spaces that generalize the classes $C_{ho}$ of Nagy and Foias. For $(ho_n)_n$ a sequence of non-zero complex numbers, we define the class $C_{(ho_n)}(H)$ as the set of operators $T in mathcal{L}(H)$ that are said to possess a $(ho_n)$-dilation: there exists a Hilbert space K and a unitary operator $U in mathcal{L}(K)$ with $H subset K$ and $T^n=ho_n P_H U^n|_H$ for every $n geq 1$ ($P_H in mathcal{L}(K)$ being the orthogonal projection from K onto its closed subspace H). These classes can be associated with an holomorphic map $f_{(ho_n)}$ as well as a quasi-norm $w_{(ho_n)}$. These three objects are tied together and we use them to characterize, describe, and give several spectral properties of operators belonging to this class.We give multiple relationships between multiple classes of this form, generalize many results that were known for classes $C_{(ho)}$, and give several examples and cases that exhibit new behaviours. We also bring a new geometric meaning behind a relationship between quasi-norms $w_{ho}$ and extend the computations of $w_{ho}(T)$ for operators T that are zeroes of a degree two polynomial.- The second main part of our study concerns classes of L^p-projections.An L^p-projection on a Banach space X, for $1leq p leq +infty$, is an idempotent operator P satisfying $ |f|_X = |(|P(f)|_X, |(I-P)(f)|_X) |_{ell_{p}}$ for all f in X. This is anL^p version of the equality $|f|^2=|Q(f)|^2 + |(I-Q)(f)|^2$, valid for orthogonal projections on Hilbert spaces.We are interested into relationships between L^p-projections on a Banach space X and L^p-projections on a subspace F, on a quotient X/F, or on a subspace of a quotient G/F. These questions are given an answer on Banach spaces with additional properties, depending on the value of p.We also introduce a notion of maximal L^p-projections for X, that is L^p-projections defined on a subspace G of X that cannot be extended to L^p-projections on larger subspaces, and study their properties, especially on finite dimensional Banach spaces. A characterization of L^{infty}-projections on every space L^{infty}(Omega) is obtained as well using new methods, generalizing previously known results

This book explores ephemeral exhibition spaces between 1750 and 1918. The chapters focus on two related spaces: the domestic interior and its imagery, and exhibitions and museums that display both national/imperial identity and the otherness that lurks beyond a country’s borders. What is revealed is that the same tension operates in these private and public realms; namely, that between identification and self-projection, on the one hand, and alienation, otherness and objectification on the other. In uncovering this, the authors show that the self, the citizen/society and the other are realities that are constantly being asserted, defined and objectified. This takes place, they demonstrate, in a ceaseless dynamic of projection versus alienation, and intimacy versus distancing.

AbstractDeveloping systems that foster situation awareness in operators requires that stakeholders can make informed decisions about the design. These decisions must account for the operator’s underlying cognitive processes based on perception, comprehension, and projection of the system state. This chapter reviews the core cognitive processes responsible for monitoring and responding to changes in system state. Operators must perceive information before they can act in response, and the interface design affects operator accuracy and speed via known mechanisms (i.e., effects of color on visual search time). Perception of key information also relies on how the operator thinks during tasks, and certain design choices can support better attention control and detection of signals. After perceiving the information, operators also must comprehend and interpret the information. Design guidance and factors related to supporting comprehension are presented alongside explanations of how cognitive load and working memory affect the operator’s ability to develop and maintain a useful mental model of the system. This review of cognitive mechanisms gives designers a strong foundation to make informed decisions ranging from choosing an alarm color to assessing how much information should be on screen at once.

1.1 Macroeconomic summary Economic recovery has consistently outperformed the technical staff’s expectations following a steep decline in activity in the second quarter of 2020. At the same time, total and core inflation rates have fallen and remain at low levels, suggesting that a significant element of the reactivation of Colombia’s economy has been related to recovery in potential GDP. This would support the technical staff’s diagnosis of weak aggregate demand and ample excess capacity. The most recently available data on 2020 growth suggests a contraction in economic activity of 6.8%, lower than estimates from January’s Monetary Policy Report (-7.2%). High-frequency indicators suggest that economic performance was significantly more dynamic than expected in January, despite mobility restrictions and quarantine measures. This has also come amid declines in total and core inflation, the latter of which was below January projections if controlling for certain relative price changes. This suggests that the unexpected strength of recent growth contains elements of demand, and that excess capacity, while significant, could be lower than previously estimated. Nevertheless, uncertainty over the measurement of excess capacity continues to be unusually high and marked both by variations in the way different economic sectors and spending components have been affected by the pandemic, and by uneven price behavior. The size of excess capacity, and in particular the evolution of the pandemic in forthcoming quarters, constitute substantial risks to the macroeconomic forecast presented in this report. Despite the unexpected strength of the recovery, the technical staff continues to project ample excess capacity that is expected to remain on the forecast horizon, alongside core inflation that will likely remain below the target. Domestic demand remains below 2019 levels amid unusually significant uncertainty over the size of excess capacity in the economy. High national unemployment (14.6% for February 2021) reflects a loose labor market, while observed total and core inflation continue to be below 2%. Inflationary pressures from the exchange rate are expected to continue to be low, with relatively little pass-through on inflation. This would be compatible with a negative output gap. Excess productive capacity and the expectation of core inflation below the 3% target on the forecast horizon provide a basis for an expansive monetary policy posture. The technical staff’s assessment of certain shocks and their expected effects on the economy, as well as the presence of several sources of uncertainty and related assumptions about their potential macroeconomic impacts, remain a feature of this report. The coronavirus pandemic, in particular, continues to affect the public health environment, and the reopening of Colombia’s economy remains incomplete. The technical staff’s assessment is that the COVID-19 shock has affected both aggregate demand and supply, but that the impact on demand has been deeper and more persistent. Given this persistence, the central forecast accounts for a gradual tightening of the output gap in the absence of new waves of contagion, and as vaccination campaigns progress. The central forecast continues to include an expected increase of total and core inflation rates in the second quarter of 2021, alongside the lapse of the temporary price relief measures put in place in 2020. Additional COVID-19 outbreaks (of uncertain duration and intensity) represent a significant risk factor that could affect these projections. Additionally, the forecast continues to include an upward trend in sovereign risk premiums, reflected by higher levels of public debt that in the wake of the pandemic are likely to persist on the forecast horizon, even in the context of a fiscal adjustment. At the same time, the projection accounts for the shortterm effects on private domestic demand from a fiscal adjustment along the lines of the one currently being proposed by the national government. This would be compatible with a gradual recovery of private domestic demand in 2022. The size and characteristics of the fiscal adjustment that is ultimately implemented, as well as the corresponding market response, represent another source of forecast uncertainty. Newly available information offers evidence of the potential for significant changes to the macroeconomic scenario, though without altering the general diagnosis described above. The most recent data on inflation, growth, fiscal policy, and international financial conditions suggests a more dynamic economy than previously expected. However, a third wave of the pandemic has delayed the re-opening of Colombia’s economy and brought with it a deceleration in economic activity. Detailed descriptions of these considerations and subsequent changes to the macroeconomic forecast are presented below. The expected annual decline in GDP (-0.3%) in the first quarter of 2021 appears to have been less pronounced than projected in January (-4.8%). Partial closures in January to address a second wave of COVID-19 appear to have had a less significant negative impact on the economy than previously estimated. This is reflected in figures related to mobility, energy demand, industry and retail sales, foreign trade, commercial transactions from selected banks, and the national statistics agency’s (DANE) economic tracking indicator (ISE). Output is now expected to have declined annually in the first quarter by 0.3%. Private consumption likely continued to recover, registering levels somewhat above those from the previous year, while public consumption likely increased significantly. While a recovery in investment in both housing and in other buildings and structures is expected, overall investment levels in this case likely continued to be low, and gross fixed capital formation is expected to continue to show significant annual declines. Imports likely recovered to again outpace exports, though both are expected to register significant annual declines. Economic activity that outpaced projections, an increase in oil prices and other export products, and an expected increase in public spending this year account for the upward revision to the 2021 growth forecast (from 4.6% with a range between 2% and 6% in January, to 6.0% with a range between 3% and 7% in April). As a result, the output gap is expected to be smaller and to tighten more rapidly than projected in the previous report, though it is still expected to remain in negative territory on the forecast horizon. Wide forecast intervals reflect the fact that the future evolution of the COVID-19 pandemic remains a significant source of uncertainty on these projections. The delay in the recovery of economic activity as a result of the resurgence of COVID-19 in the first quarter appears to have been less significant than projected in the January report. The central forecast scenario expects this improved performance to continue in 2021 alongside increased consumer and business confidence. Low real interest rates and an active credit supply would also support this dynamic, and the overall conditions would be expected to spur a recovery in consumption and investment. Increased growth in public spending and public works based on the national government’s spending plan (Plan Financiero del Gobierno) are other factors to consider. Additionally, an expected recovery in global demand and higher projected prices for oil and coffee would further contribute to improved external revenues and would favor investment, in particular in the oil sector. Given the above, the technical staff’s 2021 growth forecast has been revised upward from 4.6% in January (range from 2% to 6%) to 6.0% in April (range from 3% to 7%). These projections account for the potential for the third wave of COVID-19 to have a larger and more persistent effect on the economy than the previous wave, while also supposing that there will not be any additional significant waves of the pandemic and that mobility restrictions will be relaxed as a result. Economic growth in 2022 is expected to be 3%, with a range between 1% and 5%. This figure would be lower than projected in the January report (3.6% with a range between 2% and 6%), due to a higher base of comparison given the upward revision to expected GDP in 2021. This forecast also takes into account the likely effects on private demand of a fiscal adjustment of the size currently being proposed by the national government, and which would come into effect in 2022. Excess in productive capacity is now expected to be lower than estimated in January but continues to be significant and affected by high levels of uncertainty, as reflected in the wide forecast intervals. The possibility of new waves of the virus (of uncertain intensity and duration) represents a significant downward risk to projected GDP growth, and is signaled by the lower limits of the ranges provided in this report. Inflation (1.51%) and inflation excluding food and regulated items (0.94%) declined in March compared to December, continuing below the 3% target. The decline in inflation in this period was below projections, explained in large part by unanticipated increases in the costs of certain foods (3.92%) and regulated items (1.52%). An increase in international food and shipping prices, increased foreign demand for beef, and specific upward pressures on perishable food supplies appear to explain a lower-than-expected deceleration in the consumer price index (CPI) for foods. An unexpected increase in regulated items prices came amid unanticipated increases in international fuel prices, on some utilities rates, and for regulated education prices. The decline in annual inflation excluding food and regulated items between December and March was in line with projections from January, though this included downward pressure from a significant reduction in telecommunications rates due to the imminent entry of a new operator. When controlling for the effects of this relative price change, inflation excluding food and regulated items exceeds levels forecast in the previous report. Within this indicator of core inflation, the CPI for goods (1.05%) accelerated due to a reversion of the effects of the VAT-free day in November, which was largely accounted for in February, and possibly by the transmission of a recent depreciation of the peso on domestic prices for certain items (electric and household appliances). For their part, services prices decelerated and showed the lowest rate of annual growth (0.89%) among the large consumer baskets in the CPI. Within the services basket, the annual change in rental prices continued to decline, while those services that continue to experience the most significant restrictions on returning to normal operations (tourism, cinemas, nightlife, etc.) continued to register significant price declines. As previously mentioned, telephone rates also fell significantly due to increased competition in the market. Total inflation is expected to continue to be affected by ample excesses in productive capacity for the remainder of 2021 and 2022, though less so than projected in January. As a result, convergence to the inflation target is now expected to be somewhat faster than estimated in the previous report, assuming the absence of significant additional outbreaks of COVID-19. The technical staff’s year-end inflation projections for 2021 and 2022 have increased, suggesting figures around 3% due largely to variation in food and regulated items prices. The projection for inflation excluding food and regulated items also increased, but remains below 3%. Price relief measures on indirect taxes implemented in 2020 are expected to lapse in the second quarter of 2021, generating a one-off effect on prices and temporarily affecting inflation excluding food and regulated items. However, indexation to low levels of past inflation, weak demand, and ample excess productive capacity are expected to keep core inflation below the target, near 2.3% at the end of 2021 (previously 2.1%). The reversion in 2021 of the effects of some price relief measures on utility rates from 2020 should lead to an increase in the CPI for regulated items in the second half of this year. Annual price changes are now expected to be higher than estimated in the January report due to an increased expected path for fuel prices and unanticipated increases in regulated education prices. The projection for the CPI for foods has increased compared to the previous report, taking into account certain factors that were not anticipated in January (a less favorable agricultural cycle, increased pressure from international prices, and transport costs). Given the above, year-end annual inflation for 2021 and 2022 is now expected to be 3% and 2.8%, respectively, which would be above projections from January (2.3% and 2,7%). For its part, expected inflation based on analyst surveys suggests year-end inflation in 2021 and 2022 of 2.8% and 3.1%, respectively. There remains significant uncertainty surrounding the inflation forecasts included in this report due to several factors: 1) the evolution of the pandemic; 2) the difficulty in evaluating the size and persistence of excess productive capacity; 3) the timing and manner in which price relief measures will lapse; and 4) the future behavior of food prices. Projected 2021 growth in foreign demand (4.4% to 5.2%) and the supposed average oil price (USD 53 to USD 61 per Brent benchmark barrel) were both revised upward. An increase in long-term international interest rates has been reflected in a depreciation of the peso and could result in relatively tighter external financial conditions for emerging market economies, including Colombia. Average growth among Colombia’s trade partners was greater than expected in the fourth quarter of 2020. This, together with a sizable fiscal stimulus approved in the United States and the onset of a massive global vaccination campaign, largely explains the projected increase in foreign demand growth in 2021. The resilience of the goods market in the face of global crisis and an expected normalization in international trade are additional factors. These considerations and the expected continuation of a gradual reduction of mobility restrictions abroad suggest that Colombia’s trade partners could grow on average by 5.2% in 2021 and around 3.4% in 2022. The improved prospects for global economic growth have led to an increase in current and expected oil prices. Production interruptions due to a heavy winter, reduced inventories, and increased supply restrictions instituted by producing countries have also contributed to the increase. Meanwhile, market forecasts and recent Federal Reserve pronouncements suggest that the benchmark interest rate in the U.S. will remain stable for the next two years. Nevertheless, a significant increase in public spending in the country has fostered expectations for greater growth and inflation, as well as increased uncertainty over the moment in which a normalization of monetary policy might begin. This has been reflected in an increase in long-term interest rates. In this context, emerging market economies in the region, including Colombia, have registered increases in sovereign risk premiums and long-term domestic interest rates, and a depreciation of local currencies against the dollar. Recent outbreaks of COVID-19 in several of these economies; limits on vaccine supply and the slow pace of immunization campaigns in some countries; a significant increase in public debt; and tensions between the United States and China, among other factors, all add to a high level of uncertainty surrounding interest rate spreads, external financing conditions, and the future performance of risk premiums. The impact that this environment could have on the exchange rate and on domestic financing conditions represent risks to the macroeconomic and monetary policy forecasts. Domestic financial conditions continue to favor recovery in economic activity. The transmission of reductions to the policy interest rate on credit rates has been significant. The banking portfolio continues to recover amid circumstances that have affected both the supply and demand for loans, and in which some credit risks have materialized. Preferential and ordinary commercial interest rates have fallen to a similar degree as the benchmark interest rate. As is generally the case, this transmission has come at a slower pace for consumer credit rates, and has been further delayed in the case of mortgage rates. Commercial credit levels stabilized above pre-pandemic levels in March, following an increase resulting from significant liquidity requirements for businesses in the second quarter of 2020. The consumer credit portfolio continued to recover and has now surpassed February 2020 levels, though overall growth in the portfolio remains low. At the same time, portfolio projections and default indicators have increased, and credit establishment earnings have come down. Despite this, credit disbursements continue to recover and solvency indicators remain well above regulatory minimums. 1.2 Monetary policy decision In its meetings in March and April the BDBR left the benchmark interest rate unchanged at 1.75%.

Data management systems impose structure on data via a static representation schema or data structure. Information from the data is extracted by executing queries based on predefined operators. This paradigm restricts the searchability of the data to concepts and relationships that are known or assumed to exist among the objects. While this is an effective and efficient means of retrieving simple information, we propose that such a structure severely limits the ability to derive breakthrough knowledge that exists in data under the guise of “unknown unknowns.” A dynamic system will alleviate this dependence, allowing theoretically infinite projections of the data to reveal discoverable relationships that are hidden by traditional use case-driven, static query systems. In this paper, we propose a framework for a data-responsive query algebra based on a dynamic hyperbolic surface model. Such a model could provide more intuitive access to analytics and insights from massive, aggregated datasets than existing methods. This model will significantly alter the means of addressing the underlying data by representing it as an arrangement on a dynamic, hyperbolic plane. Consequently, querying the data can be viewed as a process similar to quantum annealing, in terms of characterizing data representation as an energy minimization problem with numerous minima.