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Let $q$ be a prime power and $k$ be a natural number. What are the possible cardinalities of point sets ${S}$ in a projective plane of order $q$, which do not intersect any line at exactly $k$ points? This problem and its variants have been investigated before, in relation with blocking sets, untouchable sets or sets of even type, among others. In this paper we show a series of results which point out the existence of all or almost all possible values $m\in [0, q^2+q+1]$ for $|S|=m$, provided that $k$ is not close to the extremal values $0$ or $q+1$. Moreover, using polynomial techniques we show the existence of a point set $S$ with the following property: for every prescribed list of numbers $t_1, \ldots t_{q^2+q+1}$, $|S\cap \ell_i|\neq t_i$ holds for the $i$th line $\ell_i$, $\forall i \in \{1, 2, \ldots, q^2+q+1\}$.

It is challenging to coordinate multiple distributed energy resources in a single or multiple buildings to ensure efficient and flexible operation. Advanced control algorithms such as model predictive control and reinforcement learning control provide solutions to this problem by effectively managing a distribution of distributed energy resource control tasks while adapting to unique building characteristics, and cooperating towards improving multi-objective key performance indicator. Yet, a research gap for advanced control adoption is the ability to benchmark algorithm performance. CityLearn addresses this gap an open-source Gym environment for the easy implementation and benchmarking of simple rule-based control and advanced algorithms that has an advantage of modeling simplicity, multi-agent control, district-level objectives, and control resiliency assessment. Here we demonstrate the functionalities of CityLearn using 17 different building control problems that have varying complexity with respect to the number of controllable distributed energy resources in buildings, the simplicity of the control algorithm, the control objective, and district size., Comment: To be published in IBPSA-USA SimBuild 2024 Conference

As more distributed energy resources become part of the demand-side infrastructure, it is important to quantify the energy flexibility they provide on a community scale, particularly to understand the impact of geographic, climatic, and occupant behavioral differences on their effectiveness, as well as identify the best control strategies to accelerate their real-world adoption. CityLearn provides an environment for benchmarking simple and advanced distributed energy resource control algorithms including rule-based, model-predictive, and reinforcement learning control. CityLearn v2 presented here extends CityLearn v1 by providing a simulation environment that leverages the End-Use Load Profiles for the U.S. Building Stock dataset to create virtual grid-interactive communities for resilient, multi-agent distributed energy resources and objective control with dynamic occupant feedback. This work details the v2 environment design and provides application examples that utilize reinforcement learning to manage battery energy storage system charging/discharging cycles, vehicle-to-grid control, and thermal comfort during heat pump power modulation.

Intelligent energy management strategies, such as Vehicle-to-Grid (V2G) and Grid-to-Vehicle (G2V) emerge as a potential solution to the Electric Vehicles' (EVs) integration into the energy grid. These strategies promise enhanced grid resilience and economic benefits for both vehicle owners and grid operators. Despite the announced prospective, the adoption of these strategies is still hindered by an array of operational problems. Key among these is the lack of a simulation platform that allows to validate and refine V2G and G2V strategies. Including the development, training, and testing in the context of Energy Communities (ECs) incorporating multiple flexible energy assets. Addressing this gap, first we introduce the EVLearn, a simulation module for researching in both V2G and G2V energy management strategies, that models EVs, their charging infrastructure and associated energy flexibility dynamics; second, this paper integrates EVLearn with the existing CityLearn framework, providing V2G and G2V simulation capabilities into the study of broader energy management strategies. Results validated EVLearn and its integration into CityLearn, where the impact of these strategies is highlighted through a comparative simulation scenario., Comment: 10 pages, 7 figures, 3 tables, 11 equations

An internal or friendly partition of a vertex set $V(G)$ of a graph $G$ is a partition to two nonempty sets $A\cup B$ such that every vertex has at least as many neighbours in its own class as in the other one. Motivated by Diwan's existence proof on internal partitions of graphs with high girth, we give constructive proofs for the existence of internal partitions in the incidence graph of projective planes and discuss its geometric properties. In addition, we determine exactly the maximum possible difference between the sizes of the neighbor set in its own class and the neighbor set of the other class, that can be attained for all vertices at the same time for the incidence graphs of desarguesian planes of square order.

Let $G$ denote an Abelian group, written additively. A $3$-term arithmetic progression of $G$, $3$-$\mathrm{AP}$ for short, is a set of three distinct elements of $G$ of the form $g$, $g+d$, $g+2d$, for some $g,d \in G$. A set $S\subseteq G$ is called $3$-$\mathrm{AP}$ free if it does not contain a $3$-$\mathrm{AP}$. $S$ is called complete $3$-$\mathrm{AP}$ free if it is $3$-$\mathrm{AP}$ free and the addition of any further element would violate the $3$-$\mathrm{AP}$ free property. In case $S$ is not necessarily $3$-$\mathrm{AP}$ free but for each $x \in G \setminus S$ there is a $3$-$\mathrm{AP}$ of $G$ consisting of $x$ and two elements of $S$, we call $S$ saturating w.r.t. $3$-$\mathrm{AP}$s. A classical problem in additive combinatorics is to determine the maximal size of a $3$-$\mathrm{AP}$ free set. A classical problem in finite geometry and coding theory is to find the minimal size of a complete cap in affine spaces, where a cap is a point set without a collinear triplet and in $\mathbb{F}_3^n$ caps and $3$-$\mathrm{AP}$ free sets are the same objects. In this paper we determine the minimum size of complete $3$-$\mathrm{AP}$ free sets and $3$-$\mathrm{AP}$ saturating sets for several families of vector spaces and cyclic groups, up to a small multiplicative constant factor., Comment: Some typos are corrected, some new references are added

Urban Building Energy Modeling (UBEM) is an emerging method to investigate urban design and energy systems against the increasing energy demand at urban and neighborhood levels. However, current UBEM methods are mostly physic-based and time-consuming in multiple climate change scenarios. This work proposes CityTFT, a data-driven UBEM framework, to accurately model the energy demands in urban environments. With the empowerment of the underlying TFT framework and an augmented loss function, CityTFT could predict heating and cooling triggers in unseen climate dynamics with an F1 score of 99.98 \% while RMSE of loads of 13.57 kWh.

In order to reprove an old result of R\'edei's on the number of directions determined by a set of cardinality $p$ in $\mathbb{F}_p^2$, Somlai proved that the non-constant polynomials over the field $\mathbb{F}_p$ whose range sums are equal to $p$ are of degree at least $\frac{p-1}{2}$. Here we characterise all of these polynomials having degree exactly $\frac{p-1}{2}$, if $p$ is large enough. As a consequence, for the same set of primes we re-establish the characterisation of sets with few determined directions due to Lov\'asz and Schrijver using discrete Fourier analysis.

In a graph $G$, the $2$-neighborhood of a vertex set $X$ consists of all vertices of $G$ having at least $2$ neighbors in $X$. We say that a bipartite graph $G(A,B)$ satisfies the double Hall property if $|A|\geq2$, and every subset $X \subseteq A$ of size at least $2$ has a $2$-neighborhood of size at least $|X|$. Salia conjectured that any bipartite graph $G(A,B)$ satisfying the double Hall property contains a cycle covering $A$. Here, we prove the existence of a $2$-factor covering $A$ in any bipartite graph $G(A,B)$ satisfying the double Hall property. We also show Salia's conjecture for graphs with restricted degrees of vertices in $B$. Additionally, we prove a lower bound on the number of edges in a graph satisfying the double Hall property, and the bound is sharp up to a constant factor., Comment: minor corrections; to be published in Discrete Mathematics

Storing and streaming high dimensional data for foundation model training became a critical requirement with the rise of foundation models beyond natural language. In this paper we introduce TensorBank, a petabyte scale tensor lakehouse capable of streaming tensors from Cloud Object Store (COS) to GPU memory at wire speed based on complex relational queries. We use Hierarchical Statistical Indices (HSI) for query acceleration. Our architecture allows to directly address tensors on block level using HTTP range reads. Once in GPU memory, data can be transformed using PyTorch transforms. We provide a generic PyTorch dataset type with a corresponding dataset factory translating relational queries and requested transformations as an instance. By making use of the HSI, irrelevant blocks can be skipped without reading them as those indices contain statistics on their content at different hierarchical resolution levels. This is an opinionated architecture powered by open standards and making heavy use of open-source technology. Although, hardened for production use using geospatial-temporal data, this architecture generalizes to other use case like computer vision, computational neuroscience, biological sequence analysis and more.

Building energy prediction and management has become increasingly important in recent decades, driven by the growth of Internet of Things (IoT) devices and the availability of more energy data. However, energy data is often collected from multiple sources and can be incomplete or inconsistent, which can hinder accurate predictions and management of energy systems and limit the usefulness of the data for decision-making and research. To address this issue, past studies have focused on imputing missing gaps in energy data, including random and continuous gaps. One of the main challenges in this area is the lack of validation on a benchmark dataset with various building and meter types, making it difficult to accurately evaluate the performance of different imputation methods. Another challenge is the lack of application of state-of-the-art imputation methods for missing gaps in energy data. Contemporary image-inpainting methods, such as Partial Convolution (PConv), have been widely used in the computer vision domain and have demonstrated their effectiveness in dealing with complex missing patterns. To study whether energy data imputation can benefit from the image-based deep learning method, this study compared PConv, Convolutional neural networks (CNNs), and weekly persistence method using one of the biggest publicly available whole building energy datasets, consisting of 1479 power meters worldwide, as the benchmark. The results show that, compared to the CNN with the raw time series (1D-CNN) and the weekly persistence method, neural network models with reshaped energy data with two dimensions reduced the Mean Squared Error (MSE) by 10% to 30%. The advanced deep learning method, Partial convolution (PConv), has further reduced the MSE by 20-30% than 2D-CNN and stands out among all models.

Physics-informed machine learning (PIML) is a set of methods and tools that systematically integrate machine learning (ML) algorithms with physical constraints and abstract mathematical models developed in scientific and engineering domains. As opposed to purely data-driven methods, PIML models can be trained from additional information obtained by enforcing physical laws such as energy and mass conservation. More broadly, PIML models can include abstract properties and conditions such as stability, convexity, or invariance. The basic premise of PIML is that the integration of ML and physics can yield more effective, physically consistent, and data-efficient models. This paper aims to provide a tutorial-like overview of the recent advances in PIML for dynamical system modeling and control. Specifically, the paper covers an overview of the theory, fundamental concepts and methods, tools, and applications on topics of: 1) physics-informed learning for system identification; 2) physics-informed learning for control; 3) analysis and verification of PIML models; and 4) physics-informed digital twins. The paper is concluded with a perspective on open challenges and future research opportunities., Comment: 16 pages, 4 figures, to be published in 2023 American Control Conference (ACC)

We study the set of intersection sizes of a k-dimensional affine subspace and a point set of size m \in [0, 2^n] of the n-dimensional binary affine space AG(n,2). Following the theme of Erd\H{o}s, F\"uredi, Rothschild and T. S\'os, we partially determine which local densities in k-dimensional affine subspaces are unavoidable in all $m$-element point sets in the n-dimensional affine space. We also show constructions of point sets for which the intersection sizes with $k$-dimensional affine subspaces takes values from a set of a small size compared to 2^k. These are built up from affine subspaces and so-called subspace evasive sets. Meanwhile, we improve the best known upper bounds on subspace evasive sets and apply results concerning the canonical signed-digit (CSD) representation of numbers., Comment: some typos fixed, further reference added

We explore the features of interpolating gauge for QCD. This gauge, defined by Doust and by Baulieu and Zwanziger, interpolates between Feynman gauge or Lorenz gauge and Coulomb gauge. We argue that it could be useful for defining the splitting functions for a parton shower beyond order $\as$ or for defining the infrared subtraction terms for higher order perturbative calculations., Comment: 39 pages, 18 figures; published version; title changed

The decarbonization of buildings presents new challenges for the reliability of the electrical grid as a result of the intermittency of renewable energy sources and increase in grid load brought about by end-use electrification. To restore reliability, grid-interactive efficient buildings can provide flexibility services to the grid through demand response. Residential demand response programs are hindered by the need for manual intervention by customers. To maximize the energy flexibility potential of residential buildings, an advanced control architecture is needed. Reinforcement learning is well-suited for the control of flexible resources as it is able to adapt to unique building characteristics compared to expert systems. Yet, factors hindering the adoption of RL in real-world applications include its large data requirements for training, control security and generalizability. Here we address these challenges by proposing the MERLIN framework and using a digital twin of a real-world 17-building grid-interactive residential community in CityLearn. We show that 1) independent RL-controllers for batteries improve building and district level KPIs compared to a reference RBC by tailoring their policies to individual buildings, 2) despite unique occupant behaviours, transferring the RL policy of any one of the buildings to other buildings provides comparable performance while reducing the cost of training, 3) training RL-controllers on limited temporal data that does not capture full seasonality in occupant behaviour has little effect on performance. Although, the zero-net-energy (ZNE) condition of the buildings could be maintained or worsened as a result of controlled batteries, KPIs that are typically improved by ZNE condition (electricity price and carbon emissions) are further improved when the batteries are managed by an advanced controller., Comment: under review

The classical No-Three-In-Line problem seeks the maximum number of points that may be selected from an $n\times n$ grid while avoiding a collinear triple. The maximum is well known to be linear in $n$. Following a question of Erde, we seek to select sets of large density from the infinite grid $Z^{2}$ while avoiding a collinear triple. We show the existence of such a set which contains $\Theta(n/\log^{1+\varepsilon}n)$ points in $[1,n]^{2}$ for all $n$, where $\varepsilon>0$ is an arbitrarily small real number. We also give computational evidence suggesting that a set of lattice points may exist that has at least $n/2$ points on every large enough $n\times n$ grid., Comment: 12 pages, 3 figures

In this paper we continue the study of a natural generalization of Tur\'an's forbidden subgraph problem and the Ruzsa-Szemer\'edi problem. Let $ex_F(n,G)$ denote the maximum number of edge-disjoint copies of a fixed simple graph $F$ that can be placed on an $n$-vertex ground set without forming a subgraph $G$ whose edges are from different $F$-copies. The case when both $F$ and $G$ are triangles essentially gives back the theorem of Ruzsa and Szemer\'edi. We extend their results to the case when $F$ and $G$ are arbitrary cliques by applying a number theoretic result due to Erd\H{o}s, Frankl and R\"odl. This extension in turn decides the order of magnitude for a large family of graph pairs, which will be subquadratic, but almost quadratic. Since the linear $r$-uniform hypergraph Tur\'an problems to determine $ex_r^{lin}(n,G)$ form a class of the multicolor Tur\'an problem, following the identity $ex_r^{lin}(n,G)=ex_{K_r}(n,G)$, our results determine the linear hypergraph Tur\'an numbers of every graph of girth $3$ and for every $r$ up to a subpolynomial factor. Furthermore, when $G$ is a triangle, we settle the case $F=C_5$ and give bounds for the cases $F=C_{2k+1}$, $k\ge 3$ as well.

One can use more than one scale variable to define the family of surfaces in the space of parton splitting parameters that define the evolution of a parton shower. In an earlier paper, we developed this idea for electron-positron annihilation. Here, we use multiple scale variables for a parton shower with initial state partons. Then we need a more sophisticated analysis because the evolution of parton distribution functions must be coordinated with the parton shower evolution. We make the needed connections more precise than in our earlier work, even for the case of just one scale variable. Then we develop an example with three scale variables, which leads to advantages compared to the usual shower formulation with only one scale variable. We provide results for Drell-Yan muon pair production., Comment: 43 pages, 13 figures

In this brief Snowmass White Paper for the Theory Frontier, we argue that there have been important recent developments in the algorithms used to generate a simulated parton shower and that further progress can be achieved in the coming decade. A much more detailed exposition can be found in a corresponding White Paper of the Energy Frontier., Comment: Contribution to Snowmass 2021. Revised version has a revised author list, slightly revised text, and a few added references

Increasing urbanization puts pressure on cities to prioritize sustainable growth and avoid carbon lock-in. Available modeling frameworks fall acutely of guiding such pivotal decision-making at the local level. Financial incentives, behavioral interventions, and mandates drive sustainable technology adoption, while land-use zoning plays a critical role in carbon emissions from the built environment. Researchers typically evaluate impacts of policies top down, on a national scale, or else post-hoc on developments vis-\`a-vis different polices in the past. Such analyses cannot forecast emission pathways for specific cities, and hence cannot serve as input to local policymakers. Here, we present IMPACT pathways, from a bottom-up model with residence level granularity, that integrate technology adoption policies with zoning policies, climate change, and grid decarbonization scenarios. With the city at the heart of our analysis, we identify an emission premium for sprawling and show that adverse policy combinations exist that can exhibit rebounding emissions over time., Comment: Under review

One can use more than one scale variable to specify the family of surfaces in the space of parton splitting parameters that define the evolution of a parton shower. Considering $e^+e^-$ annihilation, we use two variables, with shower evolution following a special path in this two dimensional space. In addition, we treat in a special way the part of the splitting function that has a soft emission singularity but no collinear singularity. This leads to certain advantages compared to the usual shower formulation with only one scale variable., Comment: 24 pages, 19 figures

Building upon prior research that highlighted the need for standardizing environments for building control research, and inspired by recently introduced challenges for real life reinforcement learning control, here we propose a non-exhaustive set of nine real world challenges for reinforcement learning control in grid-interactive buildings. We argue that research in this area should be expressed in this framework in addition to providing a standardized environment for repeatability. Advanced controllers such as model predictive control and reinforcement learning (RL) control have both advantages and disadvantages that prevent them from being implemented in real world problems. Comparisons between the two are rare, and often biased. By focusing on the challenges, we can investigate the performance of the controllers under a variety of situations and generate a fair comparison. As a demonstration, we implement the offline learning challenge in CityLearn and study the impact of different levels of domain knowledge and complexity of RL algorithms. We show that the sequence of operations utilized in a rule based controller (RBC) used for offline training affects the performance of the RL agents when evaluated on a set of four energy flexibility metrics. Longer offline learning from an optimized RBC leads to improved performance in the long run. RL agents that learn from a simplified RBC risk poorer performance as the offline learning period increases. We also observe no impact on performance from information sharing amongst agents. We call for a more interdisciplinary effort of the research community to address the real world challenges, and unlock the potential of grid-interactive building, Comment: under review

HVAC systems account for a significant portion of building energy use. Nighttime setback scheduling is an energy conservation measure where cooling and heating setpoints are increased and decreased respectively during unoccupied periods with the goal of obtaining energy savings. However, knowledge of a building's real occupancy is required to maximize the success of this measure. In addition, there is the need for a scalable way to estimate energy savings potential from energy conservation measures that is not limited by building specific parameters and experimental or simulation modeling investments. Here, we propose MARTINI, a sMARt meTer drIveN estImation of occupant-derived HVAC schedules and energy savings that leverages the ubiquity of energy smart meters and WiFi infrastructure in commercial buildings. We estimate the schedules by clustering WiFi-derived occupancy profiles and, energy savings by shifting ramp-up and setback times observed in typical/measured load profiles obtained by clustering smart meter energy profiles. Our case-study results with five buildings over seven months show an average of 8.1%-10.8% (summer) and 0.2%-5.9% (fall) chilled water energy savings when HVAC system operation is aligned with occupancy. We validate our method with results from building energy performance simulation (BEPS) and find that estimated average savings of MARTINI are within 0.9%-2.4% of the BEPS predictions. In the absence of occupancy information, we can still estimate potential savings from increasing ramp-up time and decreasing setback start time. In 51 academic buildings, we find savings potentials between 1%-5%., Comment: submitted

Increasing amounts of distributed generation in distribution networks can provide both challenges and opportunities for voltage regulation across the network. Intelligent control of smart inverters and other smart building energy management systems can be leveraged to alleviate these issues. GridLearn is a multiagent reinforcement learning platform that incorporates both building energy models and power flow models to achieve grid level goals, by controlling behind-the-meter resources. This study demonstrates how multi-agent reinforcement learning can preserve building owner privacy and comfort while pursuing grid-level objectives. Building upon the CityLearn framework which considers RL for building-level goals, this work expands the framework to a network setting where grid-level goals are additionally considered. As a case study, we consider voltage regulation on the IEEE-33 bus network using controllable building loads, energy storage, and smart inverters. The results show that the RL agents nominally reduce instances of undervoltages and reduce instances of overvoltages by 34%.

We consider a natural generalisation of Tur\'an's forbidden subgraph problem and the Ruzsa-Szemer\'edi problem by studying the maximum number $ex_F(n,G)$ of edge-disjoint copies of a fixed graph $F$ can be placed on an $n$-vertex ground set without forming a subgraph $G$ whose edges are from different $F$-copies. We determine the pairs $\{F, G\}$ for which the order of magnitude of $ex_F(n,G)$ is quadratic and prove several asymptotic results using various tools from the regularity lemma and supersaturation to graph packing results.

An internal or friendly partition of a graph is a partition of the vertex set into two nonempty sets so that every vertex has at least as many neighbours in its own class as in the other one. It has been shown that apart from finitely many counterexamples, every 3, 4 or 6-regular graph has an internal partition. In this note we focus on the $5$-regular case and show that among the subgraphs of minimum degree at least $3$, there are some which have small intersection. We also discuss the existence of internal partitions in some families of Cayley graphs, notably we determine all $5$-regular Abelian Cayley graphs which do not have an internal partition.