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We consider ideals in a polynomial ring generated by collections of power sum polynomials, and obtain conditions under which these define complete intersection rings, normal domains, and unique factorization domains. We also settle a key case of a conjecture of Conca, Krattenthaler, and Watanabe, and prove other results in that direction.

Inspired by the work of Fr\"oberg (1990) and Eagon and Reiner (1998), Bayer et al. recently introduced two new graph complexes: \emph{total cut complexes} and \emph{cut complexes}. In this article, we investigate these complexes specifically for (rectangular) grid graphs, focusing on $2 \times n$ and $3 \times n$ cases. We extend and refine the work of Bayer et al., proving and strengthening several of their conjectures, thereby enhancing the understanding of these graph complexes' topological and combinatorial properties., Comment: Minor changes

The perfect matching complex of a simple graph G is a simplicial complex having facets (maximal faces) as the perfect matchings of G. This article discusses the perfect matching complex of polygonal line tiling and the $\left(2 \times n\right)$-grid graph in particular. We use tools from discrete Morse theory to show that the perfect matching complex of any polygonal line tiling is either contractible or homotopically equivalent to a wedge of spheres. While proving our results, we also characterise all the matchings that can not be extended to form a perfect matching., Comment: 24 pages

Malaria remains one of the most pressing public health concerns globally, causing significant morbidity and mortality, especially in sub-Saharan Africa. Rapid and accurate diagnosis is crucial for effective treatment and disease management. Traditional diagnostic methods, such as microscopic examination of blood smears, are labor-intensive and require significant expertise, which may not be readily available in resource-limited settings. This project aims to automate the detection of malaria-infected cells using a deep learning approach. We employed a convolutional neural network (CNN) based on the ResNet50 architecture, leveraging transfer learning to enhance performance. The Malaria Cell Images Dataset from Kaggle, containing 27,558 images categorized into infected and uninfected cells, was used for training and evaluation. Our model demonstrated high accuracy, precision, and recall, indicating its potential as a reliable tool for assisting in malaria diagnosis. Additionally, a web application was developed using Streamlit to allow users to upload cell images and receive predictions about malaria infection, making the technology accessible and user-friendly. This paper provides a comprehensive overview of the methodology, experiments, and results, highlighting the effectiveness of deep learning in medical image analysis.

For $K$ an infinite field of characteristic other than two, consider the action of the special orthogonal group $\operatorname{SO}_t(K)$ on a polynomial ring via copies of the regular representation. When $K$ has characteristic zero, Boutot's theorem implies that the invariant ring has rational singularities; when $K$ has positive characteristic, the invariant ring is $F$-regular, as proven by Hashimoto using good filtrations. We give a new proof of this, viewing the invariant ring for $\operatorname{SO}_t(K)$ as a cyclic cover of the invariant ring for the corresponding orthogonal group; this point of view has a number of useful consequences, for example it readily yields the $a$-invariant and information on the Hilbert series. Indeed, we use this to show that the $h$-vector of the invariant ring for $\operatorname{SO}_t(K)$ need not be unimodal.

We investigate quotients by radical monomial ideals for which $T^2$, the second cotangent cohomology module, vanishes. The dimension of the graded components of $T^2$, and thus their vanishing, depends only on the combinatorics of the corresponding simplicial complex. We give both a complete characterization and a full list of one dimensional complexes with $T^2=0$. We characterize the graded components of $T^2$ when the simplicial complex is a uniform matroid. Finally, we show that $T^2$ vanishes for all matroids of corank at most two and conjecture that all connected matroids with vanishing $T^2$ are of corank at most two., Comment: 13 pages

Zero forcing in graphs is a coloring process where a colored vertex can force its unique uncolored neighbor to be colored. A zero forcing set is a set of initially colored vertices capable of eventually coloring all vertices of the graph. In this paper, we focus on the numbers $z(G; i)$, which is the number of zero forcing sets of size $i$ of the graph $G$. These numbers were initially studied by Boyer et al. where they conjectured that for any graph $G$ on $n$ vertices, $z(G; i) \leq z(P_n; i)$ for all $i \geq 1$ where $P_n$ is the path graph on $n$ vertices. The main aim of this paper is to show that several classes of graphs, including outerplanar graphs and threshold graphs, satisfy this conjecture. We do this by studying various graph operations and examining how they affect the number of zero forcing sets., Comment: 17 pages, 17 figures

Out-of-distribution (OOD) generalisation is challenging because it involves not only learning from empirical data, but also deciding among various notions of generalisation, e.g., optimising the average-case risk, worst-case risk, or interpolations thereof. While this choice should in principle be made by the model operator like medical doctors, this information might not always be available at training time. The institutional separation between machine learners and model operators leads to arbitrary commitments to specific generalisation strategies by machine learners due to these deployment uncertainties. We introduce the Imprecise Domain Generalisation framework to mitigate this, featuring an imprecise risk optimisation that allows learners to stay imprecise by optimising against a continuous spectrum of generalisation strategies during training, and a model framework that allows operators to specify their generalisation preference at deployment. Supported by both theoretical and empirical evidence, our work showcases the benefits of integrating imprecision into domain generalisation.

In computational chemistry, accurately predicting molecular configurations that exhibit specific properties remains a critical challenge. Its intricacies become especially evident in the study of molecular aggregates, where the light-induced functionality is tied to highly structure-dependent electronic couplings between molecules. Here, we present an efficient strategy for the targeted screening of the structural space employing a "functionality optimization" technique, in which a chosen descriptor, constrained by the ground state energy expression, is optimized. The chosen algorithmic differentiation (AD) framework allows to automatically obtain gradients without its tedious implementation. We demonstrate the effectiveness of the approach by identifying Perylene Bisiimide (PBI) dimer motifs with enhanced SF rate. Our findings reveal that certain structural modifications of the PBI monomer, such as helical twisting and bending, as well as slipped-rotated packing arrangements, can significantly increase the SF rate.

With the increasing influence of social media, online misinformation has grown to become a societal issue. The motivation for our work comes from the threat caused by cheapfakes, where an unaltered image is described using a news caption in a new but false-context. The main challenge in detecting such out-of-context multimedia is the unavailability of large-scale datasets. Several detection methods employ randomly selected captions to generate out-of-context training inputs. However, these randomly matched captions are not truly representative of out-of-context scenarios due to inconsistencies between the image description and the matched caption. We aim to address these limitations by introducing a novel task of out-of-context caption generation. In this work, we propose a new method that generates a realistic out-of-context caption given visual and textual context. We also demonstrate that the semantics of the generated captions can be controlled using the textual context. We also evaluate our method against several baselines and our method improves over the image captioning baseline by 6.2% BLUE-4, 2.96% CiDEr, 11.5% ROUGE, and 7.3% METEOR

Let $G$ be a finite simple graph and let $NI(G)$ denote the closed neighborhood ideal of $G$ in a polynomial ring $R$. We show that if $G$ is a forest, then the Castelnuovo-Mumford regularity of $R/NI(G)$ is the same as the matching number of $G$, thus proving a conjecture of Sharifan and Moradi in the affirmative. We also show that the matching number of $G$ provides a lower bound for the Castelnuovo-Mumford regularity of $R/NI(G)$ for any $G$. Furthermore, we prove that, if $G$ contains a simplicial vertex, then $NI(G)$ admits a Betti splitting, and consequently, we show that the projective dimension of $R/NI(G)$ is also bounded below by the matching number of $G$, if $G$ is a forest or a unicyclic graph., Comment: Final version. To appear in the Journal of Algebraic Combinatorics

For the theoretical screening of Singlet Fission (SF) rates in molecular aggregates, commonly dimer model systems are employed. However, there is experimental evidence, that the SF process proceeds from the $^{1}$(TT) state via an triplet-triplet energy transfer process to a further intermediate: a $^{1}$(T...T) state with two non-adjacent, spin-correlated triplets, which cannot be captured by the dimer models. In this work, we extend Michl's diabatic frontier orbital model to trimer systems, for which we automatically generate the diabatic two and three-center couplings using symbolic algebra. We apply this method to study the packing dependence of the $^{1}$(T...T) formation in the perylene diimide (PDI) trimer stack. We find that efficient triplet-triplet energy transfer is facilitated by structural motifs for which also significant excimer character can be observed. Furthermore, the coupling shows a local maximum for the structural motif that has been assigned to efficient $^{1}$(TT) population. Employing second order perturbation theory, we study the interference of the individual electronic pathways that arise in the PDI trimer system, allowing us to reproduce the packing dependence of the SF rates derived from the Redfield simulations in the work by Mirjani et. al. (Phys. Chem. C 2014, 118, 26, 14192-14199).

Let $V$ be a finite rank vector space over a perfect field of characteristic $p>0$, and let $G$ be a finite subgroup of $\operatorname{GL}(V)$. If $V$ is a permutation representation of $G$, or more generally a monomial representation, we prove that the ring of invariants $(\operatorname{Sym}V)^G$ has finite Frobenius representation type. We also construct an example with $V$ a finite rank vector space over the algebraic closure of the function field ${\mathbb{F}_3}(t)$, and $G$ an elementary abelian subgroup of $\operatorname{GL}(V)$, such that the invariant ring $(\operatorname{Sym}V)^G$ does not have finite Frobenius representation type.

A celebrated theorem of Fr\"oberg gives a complete combinatorial classification of quadratic square-free monomial ideals with a linear resolution. A generalization of this theorem to higher degree square-free monomial ideals is an active area of research. The existence of a linear resolution of such ideals often depends on the field over which the polynomial ring is defined. Hence, it is too much to expect that in the higher degree case a linear resolution can be identified purely using a combinatorial feature of an associated combinatorial structure. However, some classes of ideals having linear resolutions have been identified using combinatorial structures. In the present paper, we use the notion of $r$-independence to construct an $r$-uniform hypergraph from the given graph. We then show that when the underlying graph is co-chordal, the corresponding edge ideal is vertex splittable, a condition stronger than having a linear resolution. We use this result to explicitly compute graded Betti numbers for various graph classes. Finally, we give a different proof for the existence of a linear resolution using the topological notion of $r$-collapsibility., Comment: Final version. Accepted for publication in Journal of Commutative Algebra

The goal of this paper is to explain how basic properties of perverse sheaves sometimes translate via Riemann-Hilbert correspondences (in both characteristic $0$ and characteristic $p$) to highly non-trivial properties of singularities, especially their local cohomology. Along the way, we develop a theory of perverse $\mathbf{F}_p$-sheaves on varieties in characteristic $p$, expanding on previous work by various authors, and including a strong version of the Artin vanishing theorem., Comment: Comments welcome

Test-time defenses are used to improve the robustness of deep neural networks to adversarial examples during inference. However, existing methods either require an additional trained classifier to detect and correct the adversarial samples, or perform additional complex optimization on the model parameters or the input to adapt to the adversarial samples at test-time, resulting in a significant increase in the inference time compared to the base model. In this work, we propose a novel test-time defense strategy called Robust Feature Inference (RFI) that is easy to integrate with any existing (robust) training procedure without additional test-time computation. Based on the notion of robustness of features that we present, the key idea is to project the trained models to the most robust feature space, thereby reducing the vulnerability to adversarial attacks in non-robust directions. We theoretically characterize the subspace of the eigenspectrum of the feature covariance that is the most robust for a generalized additive model. Our extensive experiments on CIFAR-10, CIFAR-100, tiny ImageNet and ImageNet datasets for several robustness benchmarks, including the state-of-the-art methods in RobustBench show that RFI improves robustness across adaptive and transfer attacks consistently. We also compare RFI with adaptive test-time defenses to demonstrate the effectiveness of our proposed approach., Comment: Published in TMLR (28 pages, 6 figures, 20 tables)

For each positive prime integer $p$ we construct a standard graded $F$-rational ring $R$, over a field $K$ of characteristic $p$, such that $R\otimes_K\overline{K}$ is not $F$-rational. By localizing we obtain a flat local homomorphism $(R, \mathfrak{m}) \to (S, \mathfrak{n})$ such that $R$ is $F$-rational, $S/\mathfrak{m} S$ is regular (in fact, a field), but $S$ is not $F$-rational. In the process we also obtain standard graded $F$-rational rings $R$ for which $R\otimes_K R$ is not $F$-rational.

We investigate the injectivity of the Frobenius map on thickenings of smooth varieties in projective space over a field of positive characteristic. We obtain uniform bounds -- i.e., independent of the characteristic -- on the thickening that ensures an injective Frobenius map when the projective variety is a smooth complete intersection or an arbitrary projective embedding of an elliptic curve. Our bounds are sharp in the case of hypersurfaces, and in the case of elliptic curves., Comment: Comments welcome!

For $K$ a field, consider a finite subgroup $G$ of $\operatorname{GL}_n(K)$ with its natural action on the polynomial ring $R:=K[x_1,\dots,x_n]$. Let $\mathfrak{n}$ denote the homogeneous maximal ideal of the ring of invariants $R^G$. We study how the local cohomology module $H^n_{\mathfrak{n}}(R^G)$ compares with $H^n_{\mathfrak{n}}(R)^G$. Various results on the $a$-invariant and on the Hilbert series of $H^n_\mathfrak{n}(R^G)$ are obtained as a consequence.

The numbers we study in this paper are of the form $B_{n, p}(k)$, which is the number of binary words of length $n$ that contain the word $p$ (as a subsequence) exactly $k$ times. Our motivation comes from the analogous study of pattern containment in permutations. In our first set of results, we obtain explicit expressions for $B_{n, p}(k)$ for small values of $k$. We then focus on words $p$ with at most $3$ runs and study the maximum number of occurrences of $p$ a word of length $n$ can have. We also study the internal zeros in the sequence $(B_{n, p}(k))_{k \geq 0}$ for fixed $n$ and discuss the unimodality and log-concavity of such sequences., Comment: 17 pages