Stig Johan Wiklund, Magnus Ytterstad, Frank Miller
Many pharmaceutical companies face concerns with the maintenance of desired revenue levels. Sales forecasts for the current portfolio of products and projects may indicate a decline in revenue as the marketed products approach patent expiry. To counteract the potential downturn in revenue, and to establish revenue growth, an in-flow of new projects into the development phases is required. In this article, we devise an approach with which the in-flow of new projects could be optimized, while adhering to the objectives and constraints set on revenue targets, budget limitations and strategic considerations on the composition of the company's portfolio.
A sheaf of modules on a site is said to be internally projective if sheaf hom with the module preserves epimorphism. In this note, we give an example showing that internally projective sheaves of abelian groups are not in general stable under base change to a slice. This shows that internal projectivity is weaker than projectivity in the internal logic of the topos, as expressed for example in terms of Shulman's stack semantics. The sheaf of groups that we use as a counterexample comes from recent work by Clausen and Scholze on light condensed sets.
Artificial intelligence (AI) systems capable of generating creative outputs are reshaping our understanding of creativity. This shift presents an opportunity for creativity researchers to reevaluate the key components of the creative process. In particular, the advanced capabilities of AI underscore the importance of studying the internal processes of creativity. This paper explores the neurobiological machinery that underlies these internal processes and describes the experiential component of creativity. It is concluded that although the products of artificial and human creativity can be similar, the internal processes are different. The paper also discusses how AI may negatively affect the internal processes of human creativity, such as the development of skills, the integration of knowledge, and the diversity of ideas.
Pseudocolimits are formal gluing constructions that combine objects in a category indexed by a pseudofunctor. When the objects are categories and the domain of the pseudofunctor is small and filtered it has been known since Exppose 6 in SGA4 that the pseudocolimit can be computed by taking the Grothendieck construction of the pseudofunctor and inverting the class of cartesian arrows with respect to the canonical fibration. This paper is a reformatted version of a MSc thesis submitted and defended at Dalhousie University in August 2022. The first part presents a set of conditions for defining an internal category of elements of a diagram of internal categories and proves it is the oplax colimit. The second part presents a set of conditions on an ambient category and an internal category with an object of weak-equivalences that allows an internal description of the axioms for a category of (right) fractions and a definition of the internal category of (right) fractions when all the conditions and axioms are satisfied. These are combined to present a suitable context for computing the pseudocolimit of a small filtered diagram of internal categories.
We construct models in which there are stationarily many structures that exhibit different variants of internal approachability at different levels. This answers a question of Foreman. We also show that the approachability property at $μ$ is consistent with having a distinction of variants of internal approachability for stationarily many $N\in[H(μ^+)]^μ$. This is obtained using a new version of Mitchell Forcing.
We study morphisms of internal locales of Grothendieck toposes externally: treating internal locales and their morphisms as sheaves and natural transformations. We characterise those morphisms of internal locales that induce surjective geometric morphisms and geometric embeddings, demonstrating that both can be computed `pointwise'. We also show that the co-frame operations on the co-frame of internal sublocales can also be computed `pointwise' too.
Given a cartesian closed category $\mathcal{V}$, we introduce an internal category of elements $\int_\mathcal{C} F$ associated to a $\mathcal{V}$-functor $F\colon \mathcal{C}^{\mathrm{op}}\to \mathcal{V}$. When $\mathcal{V}$ is extensive, we show that this internal Grothendieck construction gives an equivalence of categories between $\mathcal{V}$-functors $\mathcal{C}^{\mathrm{op}}\to \mathcal{V}$ and internal discrete fibrations over $\mathcal{C}$, which can be promoted to an equivalence of $\mathcal{V}$-categories. Using this construction, we prove a representation theorem for $\mathcal{V}$-categories, stating that a $\mathcal{V}$-functor $F\colon \mathcal{C}^{\mathrm{op}}\to \mathcal{V}$ is $\mathcal{V}$-representable if and only if its internal category of elements $\int_\mathcal{C} F$ has an internal terminal object. We further obtain a characterization formulated completely in terms of $\mathcal{V}$-categories using shifted $\mathcal{V}$-categories of elements. Moreover, in the presence of $\mathcal{V}$-tensors, we show that it is enough to consider $\mathcal{V}$-terminal objects in the underlying $\mathcal{V}$-category $\mathrm{Und}\int_\mathcal{C} F$ to test the representability of a $\mathcal{V}$-functor $F$. We apply these results to the study of weighted $\mathcal{V}$-limits, and also obtain a novel result describing weighted $\mathcal{V}$-limits as certain conical internal limits.
I introduce a new way of decomposing the evolution of the wealth distribution using a simple continuous time stochastic model, which separates the effects of mobility, savings, labor income, rates of return, demography, inheritance, and assortative mating. Based on two results from stochastic calculus, I show that this decomposition is nonparametrically identified and can be estimated based solely on repeated cross-sections of the data. I estimate it in the United States since 1962 using historical data on income, wealth, and demography. I find that the main drivers of the rise of the top 1% wealth share since the 1980s have been, in decreasing level of importance, higher savings at the top, higher rates of return on wealth (essentially in the form of capital gains), and higher labor income inequality. I then use the model to study the effects of wealth taxation. I derive simple formulas for how the tax base reacts to the net-of-tax rate in the long run, which nest insights from several existing models, and can be calibrated using estimable elasticities. In the benchmark calibration, the revenue-maximizing wealth tax rate at the top is high (around 12%), but the revenue collected from the tax is much lower than in the static case.
We develop a number of basic concepts in the theory of categories internal to an $\infty$-topos. We discuss adjunctions, limits and colimits as well as Kan extensions for internal categories, and we use these results to prove the universal property of internal presheaf categories. We furthermore construct the free cocompletion of an internal category by colimits that are indexed by an arbitrary class of diagram shapes.
As a simple technique to accelerate inference of large-scale pre-trained models, early exiting has gained much attention in the NLP community. It allows samples to exit early at internal classifiers without passing through the entire model. Most existing work usually trains the internal classifiers independently and employs an exiting strategy to decide whether or not to exit based on the confidence of the current internal classifier. However, none of these works takes full advantage of the fact that the internal classifiers are trained to solve the same task therefore can be used to construct an ensemble. In this paper, we show that a novel objective function for the training of the ensemble internal classifiers can be naturally induced from the perspective of ensemble learning and information theory. The proposed training objective consists of two terms: one for accuracy and the other for the diversity of the internal classifiers. In contrast, the objective used in prior work is exactly the accuracy term of our training objective therefore only optimizes the accuracy but not diversity. Further, we propose a simple voting-based strategy that considers predictions of all the past internal classifiers to infer the correct label and decide whether to exit. Experimental results on various NLP tasks show that our proposed objective function and voting-based strategy can achieve better accuracy-speed trade-offs.
For a vertex $u$ of a tree $T$, the leaf (internal, respectively) status of $u$ is the sum of the distances from $u$ to all leaves (internal vertices, respectively) of $T$. The minimum (maximum, respectively) leaf status of a tree $T$ is the minimum (maximum, respectively) leaf statuses of all vertices of $T$. The minimum (maximum, respectively) internal status of a tree $T$ is the minimum (maximum, respectively) internal statuses of all vertices of $T$. We give the smallest and largest values for the minimum leaf status, maximum leaf status, minimum internal status, and maximum internal status of a tree and characterize the extremal cases. We also discuss these parameters of a tree with given diameter or maximum degree.
This paper develops a dynamic internal fraud model for operational losses in retail banking. It considers public operational losses arising from internal fraud in retail banking within a group of international banks. Additionally, the model takes into account internal factors such as the ethical quality of workers and the risk controls set by bank managers. The model is validated by measuring the impact of macroeconomic indicators such as GDP growth and the corruption perception upon the severity and frequency of losses implied by the model. In general,results show that internal fraud losses are pro-cyclical, and that country specific corruption perceptions positively affects internal fraud losses. Namely, when a country is perceived to be more corrupt, retail banking in that country will feature more severe internal fraud losses.
Knowledge distillation is typically conducted by training a small model (the student) to mimic a large and cumbersome model (the teacher). The idea is to compress the knowledge from the teacher by using its output probabilities as soft-labels to optimize the student. However, when the teacher is considerably large, there is no guarantee that the internal knowledge of the teacher will be transferred into the student; even if the student closely matches the soft-labels, its internal representations may be considerably different. This internal mismatch can undermine the generalization capabilities originally intended to be transferred from the teacher to the student. In this paper, we propose to distill the internal representations of a large model such as BERT into a simplified version of it. We formulate two ways to distill such representations and various algorithms to conduct the distillation. We experiment with datasets from the GLUE benchmark and consistently show that adding knowledge distillation from internal representations is a more powerful method than only using soft-label distillation.
Inspired by recent work of Batanin and Berger on the homotopy theory of operads, a general monad-theoretic context for speaking about structures within structures is presented, and the problem of constructing the universal ambient structure containing the prescribed internal structure is studied. Following the work of Lack, these universal objects must be constructed from simplicial objects arising from our monad-theoretic framework, as certain 2-categorical colimits called codescent objects. We isolate the extra structure present on these simplicial objects which enable their codescent objects to be computed. These are the crossed internal categories of the title, and generalise the crossed simplicial groups of Loday and Fiedorowicz. The most general results of this article are concerned with how to compute such codescent objects in 2-categories of internal categories, and on isolating conditions on the monad-theoretic situation which enable these results to apply. Combined with earlier work of the author in which operads are seen as polynomial 2-monads, our results are then applied to the theory of non-symmetric, symmetric and braided operads. In particular, the well-known construction of a PROP from an operad is recovered, as an illustration of our techniques.
In practical computation with Runge--Kutta methods, the stage equations are not satisfied exactly, due to roundoff errors, algebraic solver errors, and so forth. We show by example that propagation of such errors within a single step can have catastrophic effects for otherwise practical and well-known methods. We perform a general analysis of internal error propagation, emphasizing that it depends significantly on how the method is implemented. We show that for a fixed method, essentially any set of internal stability polynomials can be obtained by modifying the implementation details. We provide bounds on the internal error amplification constants for some classes of methods with many stages, including strong stability preserving methods and extrapolation methods. These results are used to prove error bounds in the presence of roundoff or other internal errors.
An internal partition of an $n$-vertex graph $G=(V,E)$ is a partition of $V$ such that every vertex has at least as many neighbors in its own part as in the other part. It has been conjectured that every $d$-regular graph with $n>N(d)$ vertices has an internal partition. Here we prove this for $d=6$. The case $d=n-4$ is of particular interest and leads to interesting new open problems on cubic graphs. We also provide new lower bounds on $N(d)$ and find new families of graphs with no internal partitions. Weighted versions of these problems are considered as well.
We investigate the internal structure of charged black holes with a spherically symmetric model in the double-null coordinate system. Hawking radiation is considered using the S-wave approximation of semiclassical back-reaction and discharge is simulated by supplying an oppositely charged matter to the black hole. In the stage of formation, the internal structure is determined by the mass and charge of collapsing matter. When the charge-mass ratio is small, a wormhole-like internal structure is observed. The structure becomes analogous to the static limit as the ratio reaches unity. After the formation, mass inflation induces large curvature in the internal structure, which makes the structure insensitive to the late-time perturbations. The internal structure determined from the formation seems to be maintained during a substantial mass reduction. The discharge and neutralization of charged black holes is also investigated for both non-evaporating and evaporating cases. Finally, we discuss the implications of the wormhole-like structure inside of charged black holes.