http://www.rssboard.org/rss-specification 720 https://escholarship.org/uc/ucdavismath_rw/rss Recent eScholarship items from Recent Work Sun, 15 Nov 2026 06:20:00 +0000 https://escholarship.org/uc/item/8t558828 A common side effect of pharmaceutical drugs is an increased propensity for cardiac arrhythmias. Many drugs bind to cardiac ion-channels in a state-specific manner, which alters the ionic conductances in complicated ways, making it difficult to identify the mechanisms underlying pro-arrhythmic drug effects. To better understand the fundamental mechanisms underlying the diverse effects of state-dependent sodium (Na⁺) channel blockers on cellular excitability, we consider two canonical motifs of drug-ion-channel interactions and compare the effects of Na⁺ channel blockers on the rate-dependence of peak upstroke velocity, conduction velocity, and vulnerable window size. In the literature, both motifs are referred to as "guarded receptor," but here we distinguish between state-specific binding that does not alter channel gating (referred to here as "guarded receptor") and state-specific binding that blocks certain gating transitions ("gate immobilization"). For... https://escholarship.org/uc/item/8t558828 Fri, 9 Oct 2026 00:00:00 +0000 Docken, Steffen S Clancy, Colleen E Lewis, Timothy J https://escholarship.org/uc/item/9bx6g0bp The alcohol abuse level of individuals in a population as it depends on resilience and peer influence is considered in this paper. Several simple models are studied as well as an integro-difTerential equation model which is derived using coarse graining from a pre-existing discrete network system. The connection structure of the discrete system tends to be richer than that of the integro-differential equation model; however, the continuum problem can be studied analytically using traveling wave, perturbation and phase plane techniques. The analysis presented in this paper suggests that, in both the discrete network and integro-differential models, nearly alcoholic or highly sober individuals are relatively unaffected by peer pressure, and this aspect of the models leads to an inertia in the spread of alcohol abuse or sobriety depending on the connectivity, initial conditions and resilience of the population. A related but different model is introduced that avoids this inertia.... https://escholarship.org/uc/item/9bx6g0bp Thu, 8 Oct 2026 00:00:00 +0000 French, Donald A Teymuroglu, Zeynep Lewis, Timothy J Braun, Richard J https://escholarship.org/uc/item/9b90j1nr Neuronal networks can produce stable oscillations and synchrony that are under tight control yet flexible enough to rapidly switch between dynamical states. The pacemaker nucleus in the weakly electric fish comprises a network of electrically coupled neurons that fire synchronously at high frequency. This activity sets the timing for an oscillating electric organ discharge with the lowest cycle-to-cycle variability of all known biological oscillators. Despite this high temporal precision, pacemaker activity is behaviorally modulated on millisecond time-scales for the generation of electrocommunication signals. The network mechanisms that allow for this combination of stability and flexibility are not well understood. In this study, we use an in vitro pacemaker preparation from Apteronotus leptorhynchus to characterize the neural responses elicited by the synaptic inputs underlying electrocommunication. These responses involve a variable increase in firing frequency and a prominent... https://escholarship.org/uc/item/9b90j1nr Thu, 8 Oct 2026 00:00:00 +0000 Lucas, Kathleen M Warrington, Julie Lewis, Timothy J Lewis, John E https://escholarship.org/uc/item/87p4d3qz When propagated action potentials in cardiac tissue interact with local heterogeneities, reflected waves can sometimes be induced. These reflected waves have been associated with the onset of cardiac arrhythmias, and while their generation is not well understood, their existence is linked to that of one-dimensional (1D) spiral waves. Thus, understanding the existence and stability of 1D spirals plays a crucial role in determining the likelihood of the unwanted reflected pulses. Mathematically, we probe these issues by viewing the 1D spiral as a time-periodic antisymmetric source defect. Through a combination of direct numerical simulation and continuation methods, we investigate the existence and stability of a 1D spiral wave in a qualitative ionic model to determine how the system's propensity for reflections is influenced by system parameters. Our results support and extend a previous hypothesis that the 1D spiral is an unstable periodic orbit that emerges through a global rearrangement... https://escholarship.org/uc/item/87p4d3qz Thu, 8 Oct 2026 00:00:00 +0000 Dodson, Stephanie Lewis, Timothy J https://escholarship.org/uc/item/86v89707 In cardiac tissue, the sinoatrial node (SAN) is responsible for initiating the periodic electrical pulses underlying heart beats. However, other heterogeneities (e.g., ischemic regions) can act as rogue pacemakers and produce oscillations in neighboring tissue that compete with the natural pacemaking of the SAN and cause potentially life-threatening arrhythmia. Thus, it is important to understand the physiological conditions that enable local regions of tissue to form pathological rhythms. It is well known that a small heterogeneity (a source) should not be able to easily activate a large area of excitable tissue (a sink). On a local level, this source-sink balance implies that positive curvature of a pacemaking region reduces the source-sink ratio and the ability to drive the neighboring tissue. However, while numerous studies provide evidence that supports the source-sink balance relationship, other studies have shown that for some depolarized heterogeneities, oscillations preferentially... https://escholarship.org/uc/item/86v89707 Thu, 8 Oct 2026 00:00:00 +0000 Dodson, Stephanie Meyer, Emily E Lewis, Timothy J https://escholarship.org/uc/item/73h1g440 1/f alpha power spectrum of the QRS complex does not imply fractal activation of the ventricles https://escholarship.org/uc/item/73h1g440 Thu, 8 Oct 2026 00:00:00 +0000 Lewis, TJ Guevara, MR https://escholarship.org/uc/item/6md8m010 Abstract.In models of working memory, transient stimuli are encoded by feature-selective persistent neural activity. Network models of working memory are also implicitly bistable. In the absence of a brief stimulus, only spontaneous, low-level, and presumably nonpatterned neural activity is seen. In many working-memory models, local recurrent excitation combined with long-range inhibition (Mexican hat coupling) can result in a network-induced, spatially localized persistent activity or “bump state” that coexists with a stable uniform state. There is now renewed interest in the concept that individual neurons might have some intrinsic ability to sustain persistent activity without recurrent network interactions. A recent visuospatial working-memory model (Camperi and Wang 1998) incorporates both intrinsic bistability of individual neurons within a firing rate network model and a single population of neurons on a ring with lateral inhibitory coupling. We have explored this model... https://escholarship.org/uc/item/6md8m010 Thu, 8 Oct 2026 00:00:00 +0000 Fall, Christopher P Lewis, Timothy J Rinzel, John https://escholarship.org/uc/item/6ks556xp Many neuronal circuits driving coordinated locomotion are composed of chains of half-center oscillators (HCOs) of various lengths. The HCO is a common motif in central pattern generating circuits (CPGs); an HCO consists of two neurons, or two neuronal populations, connected by reciprocal inhibition. To maintain appropriate motor coordination for effective locomotion over a broad range of frequencies, chains of CPGs must produce approximately constant phase-differences in a robust manner. In this article, we study phase-locking in chains of nearest-neighbor coupled HCOs and examine how the circuit architecture can promote phase-constancy, i.e., inter-HCO phase-differences that are frequency-invariant. We use two models with different levels of abstraction: (1) a conductance-based model in which each neuron is modeled by the Morris–Lecar equations (the ML-HCO model); and (2) a coupled phase model in which the state of each HCO is captured by its phase (the phase-HCO model). We show... https://escholarship.org/uc/item/6ks556xp Thu, 8 Oct 2026 00:00:00 +0000 Zhang, Calvin Lewis, Timothy J https://escholarship.org/uc/item/5xz99288 Models of cardiac tissue span a wide range of complexity, from minimal models to biophysically detailed models. It is often assumed that minimal models accurately capture the core behaviors of propagating action potentials in biophysically detailed models. However, it remains unclear whether additional physiological detail introduces qualitatively new behaviors. Dispersion relations quantify the existence, stability, and bifurcations of traveling waves, but their computation for biophysically detailed models has been limited by system complexity and computational cost. To overcome these challenges, we develop Traveling Wave Investigation Software Tool (TWIST), a numerical framework for the systematic computation of traveling wave solutions and their spectra. Using TWIST, we compute and compare dispersion relations in the FitzHugh-Nagumo, Beeler-Reuter, and nine-variable ten Tusscher-Panfilov models. Our analysis shows that increasing physiological detail can introduce substantial... https://escholarship.org/uc/item/5xz99288 Thu, 8 Oct 2026 00:00:00 +0000 Lovero, Vincent Dodson, Stephanie Lewis, Timothy J https://escholarship.org/uc/item/5w99v92r Recent evidence suggests that electrical coupling plays a role in generating oscillatory behaviour in networks of neurons; however, the underlying mechanisms have not been identified. Using a cellular automata model proposed by Traub et al (Traub R D, Schmitz D, Jefferys J G and Draguhn A 1999 High-frequency population oscillations are predicted to occur in hippocampal pyramidal neural networks interconnected by axo-axonal gap junctions Neuroscience 92 407-26), we describe a novel mechanism for self-organized oscillations in networks that have strong, sparse random electrical coupling via gap junctions. The network activity is generated by random spontaneous activity that is moulded into regular population oscillations by the propagation of activity through the network. We explain how this activity gives rise to particular dependences of mean oscillation frequency on network connectivity parameters and on the rate of spontaneous activity, and we derive analytical expressions to... https://escholarship.org/uc/item/5w99v92r Thu, 8 Oct 2026 00:00:00 +0000 Lewis, TJ Rinzel, J https://escholarship.org/uc/item/59s669cw Mechanism for cAMP Overshoot in Ventricular Myocytes Following β1-Adrenergic Stimulation https://escholarship.org/uc/item/59s669cw Thu, 8 Oct 2026 00:00:00 +0000 Meyer, Emily E Lewis, Timothy J Clancy, Colleen E https://escholarship.org/uc/item/4km773jd We examine how the location of weak electrical coupling affects phase-locking in a pair of model fast-spiking interneurons. Each model neuron consists of a somatic compartment and a passive dendritic compartment. At relatively low frequencies, the phase-locking structure for somatic and dendritic coupling is qualitatively the same: below a critical frequency, stable synchronous and anti-phase activity co-exist, and only synchrony is stable above this critical frequency. At higher frequencies, the synchronous state remains stable for somatic coupling, but for dendritic coupling, the synchronous state becomes unstable and anti-phase oscillations become stable. https://escholarship.org/uc/item/4km773jd Thu, 8 Oct 2026 00:00:00 +0000 Lewis, Timothy J Rinzel, John https://escholarship.org/uc/item/4335s3b7 It has been suggested that the normal irregular beating of the heart is a manifestation of deterministically chaotic dynamics. Evidence proffered in support of this hypothesis includes a 1/f-like power spectrum, a small noninteger correlation dimension, and self-similarity of the time series. The major cause of the normal fluctuations in heart rate is the impingement of several neural and hormonal control systems upon the sinoatrial node, the natural pacemaker of the heart. However, intrinsic fluctuations of beat rate can be seen in the isolated node, devoid of all neural and hormonal inputs, and even in a single cell isolated from the node. The electrical activity in such a single cell is generated by ions flowing through discrete channels in the cell membrane.We decided to test the hypothesis that the fluctuations in beat rate in a single cell might be due to the fluctuations in the activity of this population of single channels. We thus assemble a model consisting of 6000 channels... https://escholarship.org/uc/item/4335s3b7 Thu, 8 Oct 2026 00:00:00 +0000 Guevara, Michael R Lewis, Timothy J https://escholarship.org/uc/item/36h2g8rg We study the dynamics of a pair of intrinsically oscillating leaky integrate-and-fire neurons (identical and noise-free) connected by combinations of electrical and inhibitory coupling. We use the theory of weakly coupled oscillators to examine how synchronization patterns are influenced by cellular properties (intrinsic frequency and the strength of spikes) and coupling parameters (speed of synapses and coupling strengths). We find that, when inhibitory synapses are fast and the electrotonic effect of the suprathreshold portion of the spike is large, increasing the strength of weak electrical coupling promotes synchrony. Conversely, when inhibitory synapses are slow and the electrotonic effect of the suprathreshold portion of the spike is small, increasing the strength of weak electrical coupling promotes antisynchrony (see Fig. 10). Furthermore, our results indicate that, given a fixed total coupling strength, either electrical coupling alone or inhibition alone is better at... https://escholarship.org/uc/item/36h2g8rg Thu, 8 Oct 2026 00:00:00 +0000 Lewis, Timothy J Rinzel, John https://escholarship.org/uc/item/2cp09343 We study propagation failure using the one-dimensional scalar bistable equation with a passive "gap" region. By applying comparison principles for this type of equation, the problem of finding conditions for block is reduced to finding conditions for the existence of steady state solutions. We present a geometrical method that allows one to easily compute the critical gap length above which a steady state solution, and thus block, first occurs. The method also helps to uncover the general bifurcation structure of the problem including the stability of the steady state solutions. In obtaining these results, we characterize the relationship between the properties of the system and propagation failure. The method can easily be extended to other gap dynamics. We use it to show that block associated with any local inhomogeneity must be associated with a limit point bifurcation. https://escholarship.org/uc/item/2cp09343 Thu, 8 Oct 2026 00:00:00 +0000 Keener, James P Lewis, Timothy J https://escholarship.org/uc/item/2595b11n We demonstrate that single-variable integrate-and-fire models can quantitatively capture the dynamics of a physiologically detailed model for fast-spiking cortical neurons. Through a systematic set of approximations, we reduce the conductance-based model to 2 variants of integrate-and-fire models. In the first variant (nonlinear integrate-and-fire model), parameters depend on the instantaneous membrane potential, whereas in the second variant, they depend on the time elapsed since the last spike [Spike Response Model (SRM)]. The direct reduction links features of the simple models to biophysical features of the full conductance-based model. To quantitatively test the predictive power of the SRM and of the nonlinear integrate-and-fire model, we compare spike trains in the simple models to those in the full conductance-based model when the models are subjected to identical randomly fluctuating input. For random current input, the simple models reproduce 70-80 percent of the spikes... https://escholarship.org/uc/item/2595b11n Thu, 8 Oct 2026 00:00:00 +0000 Jolivet, Renaud Lewis, Timothy J Gerstner, Wulfram https://escholarship.org/uc/item/1zw589rn During forward swimming, crayfish and other long-tailed crustaceans rhythmically move four pairs of limbs called swimmerets to propel themselves through the water. This behavior is characterized by a particular stroke pattern in which the most posterior limb pair leads the rhythmic cycle and adjacent swimmerets paddle sequentially with a delay of roughly 25% of the period. The neural circuit underlying limb coordination consists of a chain of local modules, each of which controls a pair of limbs. All modules are directly coupled to one another, but the inter-module coupling strengths decrease with the distance of the connection. Prior modeling studies of the swimmeret neural circuit have included only the dominant nearest-neighbor coupling. Here, we investigate the potential modulatory role of long-range connections between modules. Numerical simulations and analytical arguments show that these connections cause decreases in the phase-differences between neighboring modules. Combined... https://escholarship.org/uc/item/1zw589rn Thu, 8 Oct 2026 00:00:00 +0000 Spardy, Lucy E Lewis, Timothy J https://escholarship.org/uc/item/0mq3x4nr We simulate the effect of periodic stimulation on a strand of ventricular muscle by numerically integrating the one-dimensional cable equation using the Beeler-Reuter model to represent the transmembrane currents. As stimulation frequency is increased, the rhythms of synchronization [1:1----2:2----2:1----4:2---- irregular----3:1----6:2----irregular----4:1----8:2----...----1:0] are successively encountered. We show that this sequence of rhythms can be accounted for by considering the response of the strand to premature stimulation. This involves deriving a one-dimensional finite-difference equation or "map" from the response to premature stimulation, and then iterating this map to predict the response to periodic stimulation. There is good quantitative agreement between the results of iteration of the map and the results of the numerical integration of the cable equation. Calculation of the Lyapunov exponent of the map yields a positive value, indicating sensitive dependence on... https://escholarship.org/uc/item/0mq3x4nr Thu, 8 Oct 2026 00:00:00 +0000 Lewis, Timothy J Guevara, Michael R https://escholarship.org/uc/item/0bz1d6h7 Recent evidence suggests that electrical coupling could play a role in generating oscillatory behavior in networks of neurons, however, exact mechanisms have not been identified. Using a cellular automata model, we recently showed that a self-organizing process can generate regular population oscillations in a network with random spontaneous activity and random gap junction-like coupling. The network activity underlying the oscillations is topologically similar to target-pattern activity. Here, we show the process at work in a biophysical model. We demonstrate that population oscillations can also arise from reentrant behavior, but these oscillations look qualitatively different than those generated by the target-pattern-like activity. https://escholarship.org/uc/item/0bz1d6h7 Thu, 8 Oct 2026 00:00:00 +0000 Lewis, Timothy J Rinzel, John https://escholarship.org/uc/item/009033xr We demonstrate that a biphasic shock is more effective than a monophasic shock at eliminating reentrant electrical activity in an ionic model of cardiac ventricular electrical activity. This effectiveness results from early hyperpolarization that enhances the recovery of sodium inactivation, thereby enabling earlier activation of recovering cells. The effect can be seen easily in a model of a single cell and also in a cable model with a ring of excitable cells. Finally, we demonstrate the phenomenon in a two-dimensional model of cardiac tissue. https://escholarship.org/uc/item/009033xr Thu, 8 Oct 2026 00:00:00 +0000 Keener, JP Lewis, TJ https://escholarship.org/uc/item/2db2p1k7 The Enhancing NeuroImaging Genetics through Meta-Analysis (ENIGMA) Consortium is a collaborative network of researchers working together on a range of large-scale studies that integrate data from 70 institutions worldwide. Organized into Working Groups that tackle questions in neuroscience, genetics, and medicine, ENIGMA studies have analyzed neuroimaging data from over 12,826 subjects. In addition, data from 12,171 individuals were provided by the CHARGE consortium for replication of findings, in a total of 24,997 subjects. By meta-analyzing results from many sites, ENIGMA has detected factors that affect the brain that no individual site could detect on its own, and that require larger numbers of subjects than any individual neuroimaging study has currently collected. ENIGMA’s first project was a genome-wide association study identifying common variants in the genome associated with hippocampal volume or intracranial volume. Continuing work is exploring genetic associations... https://escholarship.org/uc/item/2db2p1k7 Tue, 29 Sep 2026 00:00:00 +0000 Thompson, Paul M Stein, Jason L Medland, Sarah E Hibar, Derrek P Vasquez, Alejandro Arias Renteria, Miguel E Toro, Roberto Jahanshad, Neda Schumann, Gunter Franke, Barbara Wright, Margaret J Martin, Nicholas G Agartz, Ingrid Alda, Martin Alhusaini, Saud Almasy, Laura Almeida, Jorge Alpert, Kathryn Andreasen, Nancy C Andreassen, Ole A Apostolova, Liana G Appel, Katja Armstrong, Nicola J Aribisala, Benjamin Bastin, Mark E Bauer, Michael Bearden, Carrie E Bergmann, Ørjan Binder, Elisabeth B Blangero, John Bockholt, Henry J Bøen, Erlend Bois, Catherine Boomsma, Dorret I Booth, Tom Bowman, Ian J Bralten, Janita Brouwer, Rachel M Brunner, Han G Brohawn, David G Buckner, Randy L Buitelaar, Jan Bulayeva, Kazima Bustillo, Juan R Calhoun, Vince D Cannon, Dara M Cantor, Rita M Carless, Melanie A Caseras, Xavier Cavalleri, Gianpiero L Chakravarty, M Mallar Chang, Kiki D Ching, Christopher RK Christoforou, Andrea Cichon, Sven Clark, Vincent P Conrod, Patricia Coppola, Giovanni https://orcid.org/0000-0003-2105-1061 Crespo-Facorro, Benedicto Curran, Joanne E Czisch, Michael Deary, Ian J de Geus, Eco JC den Braber, Anouk Delvecchio, Giuseppe Depondt, Chantal de Haan, Lieuwe de Zubicaray, Greig I Dima, Danai Dimitrova, Rali Djurovic, Srdjan Dong, Hongwei Donohoe, Gary Duggirala, Ravindranath Dyer, Thomas D Ehrlich, Stefan Ekman, Carl Johan Elvsåshagen, Torbjørn Emsell, Louise Erk, Susanne Espeseth, Thomas Fagerness, Jesen Fears, Scott Fedko, Iryna Fernández, Guillén Fisher, Simon E Foroud, Tatiana Fox, Peter T Francks, Clyde Frangou, Sophia Frey, Eva Maria Frodl, Thomas Frouin, Vincent Garavan, Hugh Giddaluru, Sudheer Glahn, David C Godlewska, Beata Goldstein, Rita Z Gollub, Randy L Grabe, Hans J https://escholarship.org/uc/item/0sx117m4 Many drugs that block voltage-gated K+ channels encoded by the human ether-à-go-go-related gene (hERG) can cause long QT syndrome and life-threatening cardiac arrhythmias, yet the molecular mechanisms that determine this risk remain unclear. A process that may counteract arrhythmogenic hERG block, termed facilitation, is common to many clinically approved hERG blockers, including nifekalant, amiodarone, promethazine, imipramine, nortriptyline, haloperidol, verapamil, carvedilol, metoprolol, propranolol, quinidine, fluoxetine, and chlorpheniramine. Facilitation is an increase in hERG current, under certain conditions, due to these blockers. Here, we propose that an agonism-while-blocking mechanism underpins facilitation. We focus on nifekalant, a class III antiarrhythmic drug and exemplar hERG blocker that induces facilitation. We tested the hypothesis that nifekalant opens hERG channel gates while blocking, and that unblocking of these open-yet-blocked channels results in supranormal... https://escholarship.org/uc/item/0sx117m4 Wed, 9 Sep 2026 00:00:00 +0000 Docken, Steffen S Marquis, Matthew J https://orcid.org/0000-0002-5556-9072 Ngo, Khoa Wada, Yuumu Kita, Satomi Yarov-Yarovoy, Vladimir https://orcid.org/0000-0002-2325-4834 Clancy, Colleen E Vorobyov, Igor https://orcid.org/0000-0002-4767-5297 Lewis, Timothy J Furutani, Kazuharu Sack, Jon T https://escholarship.org/uc/item/8cz7s9c9 We introduce a type A crystal structure on decreasing factorizations on 321-avoiding elements in the 0-Hecke monoid which we call *-crystal. This crystal is a K-theoretic generalization of the crystal on decreasing factorizations in the symmetric group of the first and last author. We prove that under the residue map the *-crystal intertwines with the crystal on set-valued tableaux recently introduced by Monical, Pechenik and Scrimshaw. We also define a new insertion from decreasing factorization to pairs of semistandard Young tableaux and prove several properties, such as its relation to the Hecke insertion and the uncrowding algorithm. The new insertion also intertwines with the crystal operators. https://escholarship.org/uc/item/8cz7s9c9 Wed, 12 Aug 2026 00:00:00 +0000 Morse, J Pan, J Poh, W Schilling, A https://orcid.org/0000-0002-2601-7340 https://escholarship.org/uc/item/52428468 Richard Stanley through a crystal lens and from a random angle https://escholarship.org/uc/item/52428468 Wed, 12 Aug 2026 00:00:00 +0000 Schilling, Anne https://orcid.org/0000-0002-2601-7340 https://escholarship.org/uc/item/44d9j9n3 We consider generalizations of Schützenberger’s promotion operator on the set ℒ of linear extensions of a finite poset. This gives rise to a strongly connected graph on ℒ. In earlier work (Ayyer et al., J. Algebraic Combinatorics 39(4), 853–881 (2014)), we studied promotion-based Markov chains on these linear extensions which generalizes results on the Tsetlin library. We used the theory of ℛ-trivial monoids in an essential way to obtain explicitly the eigenvalues of the transition matrix in general when the poset is a rooted forest. We first survey these results and then present explicit bounds on the mixing time and conjecture eigenvalue formulas for more general posets. We also present a generalization of promotion to arbitrary subsets of the symmetric group. https://escholarship.org/uc/item/44d9j9n3 Wed, 12 Aug 2026 00:00:00 +0000 Ayyer, A Klee, S Schilling, A https://orcid.org/0000-0002-2601-7340 https://escholarship.org/uc/item/0sv7j1sr We give an explicit description of the image of a quantum LS path, regarded as a rational path, under the action of root operators, and show that the set of quantum LS paths is stable under the action of the root operators. As a by-product, we obtain a new proof of the fact that a projected level-zero LS path is just a quantum LS path. https://escholarship.org/uc/item/0sv7j1sr Wed, 12 Aug 2026 00:00:00 +0000 Lenart, Cristian Naito, Satoshi Sagaki, Daisuke Schilling, Anne https://orcid.org/0000-0002-2601-7340 Shimozono, Mark https://escholarship.org/uc/item/85n5c4rq Eight quick tips for data-model integration in ecology https://escholarship.org/uc/item/85n5c4rq Thu, 30 Jul 2026 00:00:00 +0000 Balstad, Laurinne J Brennan, Joe Baskett, Marissa L Berglund, Mattea K Blundell, Mei Z Bolin, Jessica A Briggs, Amy A Fisher, Mary C Heggerud, Christopher M Jarvis-Cross, Madeline Mossman, Lauren Odell, Andrea N Paige, Jennifer https://orcid.org/0000-0002-5753-0403 Pelletier, Sophia Provost, Mikaela M https://escholarship.org/uc/item/9r52d862 Many different microswimmers propel themselves using flagella that beat periodically. The shape of the flagellar beat and swimming speed have been observed to change with fluid rheology. We quantify changes in the flagellar waveforms of <i>Chlamydomonas reinhardtii</i> in response to changes in fluid viscosity using (i) shape mode analysis and (ii) a full swimmer simulation to analyse how shape changes affect the swimming speed and to explore the dimensionality of the shape space. By decomposing the gait into the time-independent mean shape and the time-varying stroke, we find that the flagellar mean shape substantially changes in response to viscosity, while the changes in the time-varying stroke are more subtle. Using the swimmer simulation, we quantify how the swimming speed is affected by the dimensionality of the flagellar shape reconstruction, and we show that the observed change in swimming speed with viscosity is explained by the variations in mean flagellar shape and... https://escholarship.org/uc/item/9r52d862 Wed, 29 Jul 2026 00:00:00 +0000 Gutierrez, Kelli E Thomases, Becca Arratia, Paulo E Guy, Robert D https://orcid.org/0000-0002-0602-7015 https://escholarship.org/uc/item/96k243c1 Metachronal paddling is a swimming strategy in which an organism oscillates sets of adjacent limbs with a constant phase lag, propagating a metachronal wave through its limbs and propelling it forward. This limb coordination strategy is utilized by swimmers across a wide range of Reynolds numbers, which suggests that this metachronal rhythm was selected for its optimality of swimming performance. In this study, we apply reinforcement learning to a swimmer at zero Reynolds number and investigate whether the learning algorithm selects this metachronal rhythm, or if other coordination patterns emerge. We design the swimmer agent with an elongated body and pairs of straight, inflexible paddles placed along the body for various fixed paddle spacings. Based on paddle spacing, the swimmer agent learns qualitatively different coordination patterns. At tight spacings, a back-to-front metachronal wave-like stroke emerges which resembles the commonly observed biological rhythm, but at wide... https://escholarship.org/uc/item/96k243c1 Wed, 29 Jul 2026 00:00:00 +0000 Bailey, Alana A Guy, Robert D https://orcid.org/0000-0002-0602-7015 https://escholarship.org/uc/item/4jr9b754 The classical problem of steady streaming induced by an oscillating object has been studied extensively, but prior work has focused almost exclusively on single-frequency oscillations, which result in symmetric, quadrupole-like flows. Here we demonstrate that dual-frequency oscillations induce asymmetric steady streaming with a non-zero net flux in a direction determined by the polarity of the oscillation-the oscillator serves as a pump. We use numerical simulations and asymptotic analysis at small amplitude to examine two-dimensional steady streaming around a cylinder, first focusing on frequency ratio 2. The computational experiments show asymmetrical streaming and pumping, i.e. net flux downstream. It is well known from asymptotic analysis that steady streaming is second order in amplitude, and we show that pumping occurs at third order. We then extend the analysis to general frequency ratios, where we give necessary conditions for pumping, and predict the order in amplitude... https://escholarship.org/uc/item/4jr9b754 Wed, 29 Jul 2026 00:00:00 +0000 Lee, HS Ristenpart, WD https://orcid.org/0000-0002-4935-6310 Guy, RD https://orcid.org/0000-0002-0602-7015 https://escholarship.org/uc/item/93w4r2xr We apply ideas from crystal theory to affine Schubert calculus and flag Gromov-Witten invariants. By defining operators on certain decompositions of elements in the type-A affine Weyl group, we produce a crystal reflecting the internal structure of Specht modules associated to permutation diagrams. We show how this crystal framework can be applied to study the product of a Schur function with a k-Schur function. Consequently, we prove that a subclass of 3-point Gromov-Witten invariants of complete flag varieties for Cⁿ enumerate the highest weight elements under these operators. https://escholarship.org/uc/item/93w4r2xr Wed, 15 Jul 2026 00:00:00 +0000 Morse, J Schilling, A https://orcid.org/0000-0002-2601-7340 https://escholarship.org/uc/item/89b1d3m4 We provide a new description of the Pieri rule of the homology of the affine Grassmannian and an affine analogue of the charge statistics in terms of bounded partitions. This makes it possible to extend the formulation of the Kostka–Foulkes polynomials in terms of solvable lattice models by Nakayashiki and Yamada to the affine setting. https://escholarship.org/uc/item/89b1d3m4 Wed, 15 Jul 2026 00:00:00 +0000 Morse, J Schilling, A https://orcid.org/0000-0002-2601-7340 https://escholarship.org/uc/item/83x4r5zb <p> Composing two representations of the general linear groups gives rise to Littlewood’s (outer) plethysm. On the level of characters, this poses the question of finding the Schur expansion of the plethysm of two Schur functions. A combinatorial interpretation for the Schur expansion coefficients of the plethysm of two Schur functions is, in general, still an open problem. We identify a proof technique of combinatorial representation theory, which we call the “ -perp trick”, and point out several examples in the literature where this idea is used. We use the -perp trick to give algorithms for computing monomial and Schur expansions of symmetric functions. In several special cases, these algorithms are more efficient than those currently implemented in SageMath . </p> https://escholarship.org/uc/item/83x4r5zb Wed, 15 Jul 2026 00:00:00 +0000 Colmenarejo, Laura Orellana, Rosa Saliola, Franco Schilling, Anne https://orcid.org/0000-0002-2601-7340 Zabrocki, Mike https://escholarship.org/uc/item/7kr251d1 We provide a characterization of the crystal bases for the quantum queer superalgebra recently introduced by Grantcharov et al. This characterization is a combination of local queer axioms generalizing Stembridge’s local axioms for crystal bases for simply-laced root systems, which were recently introduced by Assaf and Oguz, with further axioms and a new graph G characterizing the relations of the type A components of the queer crystal. We provide a counterexample to Assaf’s and Oguz’ conjecture that the local queer axioms uniquely characterize the queer supercrystal. We obtain a combinatorial description of the graph G on the type A components by providing explicit combinatorial rules for the odd queer operators on certain highest weight elements. https://escholarship.org/uc/item/7kr251d1 Wed, 15 Jul 2026 00:00:00 +0000 Gillespie, Maria Hawkes, Graham Poh, Wencin Schilling, Anne https://orcid.org/0000-0002-2601-7340 https://escholarship.org/uc/item/73q3q952 Dimension of Splines on Graphs in the Case of Degree Two and Smoothness One in Two Variables https://escholarship.org/uc/item/73q3q952 Wed, 15 Jul 2026 00:00:00 +0000 Nazir, Shaheen Schilling, Anne https://orcid.org/0000-0002-2601-7340 Tymoczko, Julianna https://escholarship.org/uc/item/6xp4p8dm We provide a crystal structure on the set of ordered multiset partitions, which recently arose in the pursuit of the Delta Conjecture. This conjecture was stated by Haglund, Remmel and Wilson as a generalization of the Shuffle Conjecture. Various statistics on ordered multiset partitions arise in the combinatorial analysis of the Delta Conjecture, one of them being the minimaj statistic, which is a variant of the major index statistic on words. Our crystal has the property that the minimaj statistic is constant on connected components of the crystal. In particular, this yields another proof of the Schur positivity of the graded Frobenius series of the generalization Rn,k due to Haglund, Rhoades and Shimozono of the coinvariant algebra Rn. The crystal structure also yields a bijective proof of the equidistributivity of the minimaj statistic with the major index statistic on ordered multiset partitions. https://escholarship.org/uc/item/6xp4p8dm Wed, 15 Jul 2026 00:00:00 +0000 Benkart, G Colmenarejo, L Harris, PE Orellana, R Panova, G Schilling, A https://orcid.org/0000-0002-2601-7340 Yip, M https://escholarship.org/uc/item/6570c945 We describe the implementation of queer supercrystals. Our code is documented with test suites and has been integrated into SageMath. Through computer explorations using this implementation, we provide a counterexample to Assaf's and Oguz' conjecture that their local queer axioms uniquely characterize the queer supercrystal. https://escholarship.org/uc/item/6570c945 Wed, 15 Jul 2026 00:00:00 +0000 Poh, W Schilling, A https://orcid.org/0000-0002-2601-7340 https://escholarship.org/uc/item/4cb1n525 Abstract Time series classification is crucial for numerous scientific and engineering applications. In this article, we present a numerically efficient, practically competitive, and theoretically rigorous classification method for distinguishing between two classes of locally stationary time series based on their time-domain, second-order characteristics. Our approach builds on the autoregressive approximation for locally stationary time series, imposes no requirement on the training sample size, and is shown to achieve zero misclassification error rate asymptotically when the underlying time series differ only mildly in their second-order characteristics. The new method is demonstrated to outperform a variety of state-of-the-art solutions, including wavelet-based, tree-based, convolution-based methods, as well as modern deep learning methods, through intensive numerical simulations and a real electroencephalography data analysis for epilepsy classification. https://escholarship.org/uc/item/4cb1n525 Fri, 10 Jul 2026 00:00:00 +0000 Qian, Chen Ding, Xiucai Li, Lexin https://escholarship.org/uc/item/80j0t30k We propose new scattering networks for signals measured on simplicial complexes, which we call Multiscale Hodge Scattering Networks (MHSNs). Our construction builds on multiscale basis dictionaries on simplicial complexes—namely, the κ-GHWT and κ-HGLET—which we recently developed for simplices of dimension κ ∈ N in a given simplicial complex by generalizing the node-based Generalized Haar–Walsh Transform (GHWT) and Hierarchical Graph Laplacian Eigen Transform (HGLET). Both the κ-GHWT and the κ-HGLET form redundant sets (i.e., dictionaries) of multiscale basis vectors and the corresponding expansion coefficients of a given signal. Our MHSNs adopt a layered structure analogous to a convolutional neural network (CNN), cascading the moments of the modulus of the dictionary coefficients. The resulting features are invariant to reordering of the simplices (i.e., node permutation of the underlying graphs). Importantly, the use of multiscale basis dictionaries in our MHSNs admits a natural... https://escholarship.org/uc/item/80j0t30k Thu, 26 Mar 2026 00:00:00 +0000 Saito, Naoki https://orcid.org/0000-0001-5234-4719 Schonsheck, Stefan C Shvarts, Eugene https://escholarship.org/uc/item/4kh0431d We propose a new method to extract discriminant and explainable features from a particular machine learning model, i.e., a combination of the scattering transform and the multiclass logistic regression. Although this model is well-known for its ability to learn various signal classes with high classification rate, it remains elusive to understand why it can generate such successful classification, mainly due to the nonlinearity of the scattering transform. In order to uncover the meaning of the scattering transform coefficients selected by the multiclass logistic regression (with the Lasso penalty), we adopt zeroth-order optimization algorithms to search an input pattern that maximizes the class probability of a class of interest given the learned model. In order to do so, it turns out that imposing sparsity and smoothness of input patterns is important. We demonstrate the effectiveness of our proposed method using a couple of synthetic time-series classification problems. https://escholarship.org/uc/item/4kh0431d Thu, 26 Mar 2026 00:00:00 +0000 Saito, Naoki https://orcid.org/0000-0001-5234-4719 Weber, David https://escholarship.org/uc/item/28t9z2kq We propose a new method to extract discriminant and explainable features from a particular machine learning model, i.e., a combination of the scattering transform and the multiclass logistic regression. Although this model is well-known for its ability to learn various signal classes with high classification rate, it remains elusive to understand why it can generate such successful classification, mainly due to the nonlinearity of the scattering transform. In order to uncover the meaning of the scattering transform coefficients selected by the multiclass logistic regression (with the Lasso penalty), we adopt zeroth-order optimization algorithms to search an input pattern that maximizes the class probability of a class of interest given the learned model. In order to do so, it turns out that imposing sparsity and smoothness of input patterns is important. We demonstrate the effectiveness of our proposed method using a couple of synthetic time-series classification problems. https://escholarship.org/uc/item/28t9z2kq Thu, 26 Mar 2026 00:00:00 +0000 Saito, Naoki https://orcid.org/0000-0001-5234-4719 Weber, David https://escholarship.org/uc/item/6rp7629z Inspired by the BCFW recurrence for tilings of the amplituhedron, we introduce the general framework of plabic tangles that utilizes plabic graphs to define rational maps between products of Grassmannians called promotions. The central conjecture of the paper is that promotion maps are quasi-cluster homomorphisms, which we prove for several classes of promotions. In order to define promotion maps, we utilize m-vector-relation configurations (m-VRCs) on plabic graphs. We relate m-VRCs to the degree (a.k.a ‘intersection number’) of the amplituhedron map on positroid varieties and characterize all plabic trees with intersection number one and their VRCs. Finally, we show that promotion maps admit an operad structure and, supported by the class of 4-mass box promotions, we point at new positivity properties for non-rational maps beyond cluster algebras. Promotion maps have important connections to the geometry and cluster structure of the amplituhedron and singularities of scattering... https://escholarship.org/uc/item/6rp7629z Wed, 11 Mar 2026 00:00:00 +0000 Even-Zohar, Chaim Parisi, Matteo Sherman-Bennett, Melissa Tessler, Ran Williams, Lauren https://escholarship.org/uc/item/4bd954vc Abstract. Refined canonical stable Grothendieck polynomials were introduced by Hwang et al. There exist two combinatorial models for these polynomials: one using hook-valued tableaux and the other using pairs of a semistandard Young tableau and (what we call) an exquisite tableau. An uncrowding algorithm on hook-valued tableaux was introduced by Pan et al. In this paper, we discover a novel connection between the two models via the uncrowding and Goulden and Greene’s jeu de taquin algorithms, using a classical result of Benkart, Sottile, and Stroomer on tableau switching. This connection reveals a symmetry of the uncrowding algorithm defined on hook-valued tableaux. As a corollary, we obtain another combinatorial model for the refined canonical stable Grothendieck polynomials in terms of biflagged tableaux, which naturally appear in the characterization of the image of the uncrowding map. https://escholarship.org/uc/item/4bd954vc Wed, 11 Mar 2026 00:00:00 +0000 Jang, Jihyeug Kim, Jang Soo Pan, Jianping Pappe, Joseph Schilling, Anne https://orcid.org/0000-0002-2601-7340 https://escholarship.org/uc/item/0vb996s2 We show that braid varieties for any complex simple algebraic group G$$G$$ are cluster varieties. This includes open Richardson varieties inside the flag variety G/B$$G/B$$. https://escholarship.org/uc/item/0vb996s2 Wed, 11 Mar 2026 00:00:00 +0000 Galashin, Pavel Lam, Thomas Sherman-Bennett, Melissa https://escholarship.org/uc/item/82c4q4tr We define a suitably tame class of singular symplectic curves in 4-manifolds, namely those whose singularities are modeled on complex curve singularities. We study the corresponding symplectic isotopy problem, with a focus on rational curves with irreducible singularities (rational cuspidal curves) in the complex projective plane. We prove that every such curve is isotopic to a complex curve in degrees up to five, and for curves with one singularity whose link is a torus knot. Classification results of symplectic isotopy classes rely on pseudo-holomorphic curves together with a symplectic version of birational geometry of log pairs and techniques from four-dimensional topology. https://escholarship.org/uc/item/82c4q4tr Wed, 25 Feb 2026 00:00:00 +0000 Golla, Marco Starkston, Laura https://escholarship.org/uc/item/3dd6v4x8 Rational cuspidal curves and symplectic fillings https://escholarship.org/uc/item/3dd6v4x8 Wed, 25 Feb 2026 00:00:00 +0000 Golla, Marco Starkston, Laura https://escholarship.org/uc/item/5cx7z609 In this self-contained paper, we present a theory of the piecewise linear minimal valid functions for the 1-row Gomory–Johnson infinite group problem. The non-extreme minimal valid functions are those that admit effective perturbations. We give a precise description of the space of these perturbations as a direct sum of certain finite- and infinite-dimensional subspaces. The infinite-dimensional subspaces have partial symmetries; to describe them, we develop a theory of inverse semigroups of partial bijections, interacting with the functional equations satisfied by the perturbations. Our paper provides the foundation for grid-free algorithms for the Gomory–Johnson model, in particular for testing extremality of piecewise linear functions whose breakpoints are rational numbers with huge denominators. https://escholarship.org/uc/item/5cx7z609 Sat, 14 Feb 2026 00:00:00 +0000 Hildebrand, Robert Köppe, Matthias Zhou, Yuan https://escholarship.org/uc/item/5c33591q Within the framework of the superadditive duality theory of integer programming, we study two types of dual-feasible functions of a single real variable (Alves et al., 2016). We introduce software that automates testing piecewise linear functions for maximality and extremality, enabling a computer-based search. We build a connection to cut-generating functions in the Gomory–Johnson and related models, complete the characterization of maximal functions, and prove analogues of the Gomory–Johnson 2-slope theorem and the Basu–Hildebrand–Molinaro approximation theorem. https://escholarship.org/uc/item/5c33591q Sat, 14 Feb 2026 00:00:00 +0000 Köppe, Matthias Wang, Jiawei https://escholarship.org/uc/item/5jq3z5r6 We investigate the semigroup of integer points inside a convex cone. We extend classical results in integer linear programming to integer conic programming. We show that the semigroup associated with nonpolyhedral cones can sometimes have a notion of finite generating set with the help of a group action. We show this is true for the cone of positive semidefinite matrices (PSD) and the second-order cone (SOC). Both cones have a finite generating set of integer points, similar in spirit to Hilbert bases, under the action of a finitely generated group. We also extend notions of total dual integrality, Gomory-Chvátal closure, and Carathéodory rank to integer points in arbitrary cones. https://escholarship.org/uc/item/5jq3z5r6 Wed, 11 Feb 2026 00:00:00 +0000 De Loera, Jesús A https://orcid.org/0000-0002-9556-1112 Marsters, Brittney Xu, Luze Zhang, Shixuan https://escholarship.org/uc/item/01b6946f Abstract We classify all contact structures on 3‐manifolds that admit a nonfree toric action, up to contactomorphism, and present them through explicit topological descriptions. Our classification is based on Lerman's classification of toric contact 3‐manifolds up to equivariant contactomorphism [Lerman, J. Symplectic Geom. 1 (2003), 785–828]. We also prove that every contact 3‐manifold with a nonfree toric action arises as the concave boundary of a toric linear plumbing over spheres inspired by Marinković et&nbsp;al. As a corollary of both results, we classify which contact 3‐manifolds arise as the concave boundary of a linear plumbing of&nbsp;spheres. https://escholarship.org/uc/item/01b6946f Wed, 28 Jan 2026 00:00:00 +0000 Marinković, Aleksandra Starkston, Laura https://escholarship.org/uc/item/795990w4 <p>In this paper, we provide novel mean value estimates for exponential sums related to the extended main conjecture of Vinogradov’s mean value theorem, by developing the Hardy-Littlewood circle method together with a refined shifting variables argument.</p> <p> Let be a natural number and . Define the exponential sum For , consider mean values of the exponential sums where we wrote . By making use of the aforementioned tools, we obtain the sharp upper bound for , for and . Furthermore, for , we obtain analogous results depending on a small cap decoupling inequality for the moment curves in . </p> https://escholarship.org/uc/item/795990w4 Sun, 7 Dec 2025 00:00:00 +0000 Oh, Changkeun Yeon, Kiseok https://escholarship.org/uc/item/47c6420j Abstract We define and study the contact cut graph which is an analogue of Hatcher and Thurston's cut graph for contact geometry, inspired by contact Heegaard splittings (Giroux, Proceedings of the International Congress of Mathematicians , Beijing, 2002; Torisu, Internat. Math. Res. Notices (2000), 441–454). We show how oriented paths in the contact cut graph correspond to Lefschetz fibrations and multisection with divides diagrams. We also give a correspondence for achiral Lefschetz fibrations. We use these correspondences to define a new invariant of Weinstein domains, the Weinstein ‐invariant , that is a symplectic analogue of the Kirby–Thompson's ‐invariant of smooth 4‐manifolds. We discuss the relation of Lefschetz stabilization with the Weinstein ‐invariant. We present topological and geometric constraints of Weinstein domains with . We also give two families of examples of multisections with divides that have arbitrarily large ‐invariant. https://escholarship.org/uc/item/47c6420j Mon, 1 Dec 2025 00:00:00 +0000 Castro, Nickolas A Islambouli, Gabriel Min, Jie Sakallı, Sümeyra Starkston, Laura Wu, Angela https://escholarship.org/uc/item/3dv5k0qb Our main result is a local-to-global principle for Morse quasigeodesics, maps and actions. As an application of our techniques we show algorithmic recognizability of Morse actions and construct Morse “Schottky subgroups” of higher-rank semisimple Lie groups via arguments not based on Tits pingpong. Our argument is purely geometric and proceeds by constructing equivariant Morse quasiisometric embeddings of trees into higher-rank symmetric spaces. https://escholarship.org/uc/item/3dv5k0qb Wed, 24 Sep 2025 00:00:00 +0000 Kapovich, Michael https://orcid.org/0000-0001-5475-8755 Leeb, Bernhard Porti, Joan https://escholarship.org/uc/item/92p0s02x The Combinatorial Theory Flip https://escholarship.org/uc/item/92p0s02x Wed, 27 Aug 2025 00:00:00 +0000 Schilling, Anne https://orcid.org/0000-0002-2601-7340 https://escholarship.org/uc/item/2rs1x74j Interview with Anne Schilling https://escholarship.org/uc/item/2rs1x74j Wed, 27 Aug 2025 00:00:00 +0000 https://escholarship.org/uc/item/7bg31125 Margolis, Rhodes and Schilling recently submitted two papers that proved that the complexity of a finite semigroup is computable. The purpose of this paper is to survey the basic results of Krohn–Rhodes complexity of finite semigroups and to outline the proof of its computability. https://escholarship.org/uc/item/7bg31125 Wed, 13 Aug 2025 00:00:00 +0000 Margolis, Stuart Rhodes, John Schilling, Anne https://orcid.org/0000-0002-2601-7340 https://escholarship.org/uc/item/5hx005cj <p> We study the interactions between toric manifolds and Weinstein handlebodies. We define a partially-centeredness condition on a Delzant polytope, which we prove ensures that the complement of a corresponding partial smoothing of the total toric divisor supports an explicit Weinstein structure. Many examples which fail this condition also fail to have Weinstein (or even exact) complement to the partially smoothed divisor. We investigate the combinatorial possibilities of Delzant polytopes that realize such Weinstein domain complements. We also develop an algorithm to construct a Weinstein handlebody diagram in Gompf standard form for the complement of such a partially smoothed total toric divisor. The algorithm we develop more generally outputs a Weinstein handlebody diagram for any Weinstein 4-manifold constructed by attaching 2-handles to the disk cotangent bundle of any surface&nbsp; , where the 2-handles are attached along the co-oriented conormal lifts of curves on&nbsp;... https://escholarship.org/uc/item/5hx005cj Thu, 17 Jul 2025 00:00:00 +0000 Acu, Bahar Capovilla-Searle, Orsola Gadbled, Agnès Marinković, Aleksandra Murphy, Emmy Starkston, Laura Wu, Angela https://escholarship.org/uc/item/5c22v7db A famous result in arithmetic Ramsey theory says that for many linear homogeneous equations E there is a threshold value Rk(E) (the Rado number of E) such that for any k-coloring of the integers in the interval [1,n], with n≥Rk(E), there exists at least one monochromatic solution. But one can further ask, how many monochromatic solutions is the minimum possible in terms of n? Several authors have estimated this function before, here we offer new tools from integer and semidefinite optimization that help find either optimal or near optimal 2-colorings minimizing the number of monochromatic solutions of several families of 3-variable non-regular homogeneous linear equations. In the last part of the paper we further extend to three and more colors for the Schur equation, improving earlier work. https://escholarship.org/uc/item/5c22v7db Sun, 25 May 2025 00:00:00 +0000 De Loera, Jesús A https://orcid.org/0000-0002-9556-1112 Ventura, Denae Wang, Liuyue Wesley, William J https://escholarship.org/uc/item/5pg5p7j7 Abstract: We study classes of domains in with sufficiently flat boundaries that admit a decomposition or covering of bounded overlap by Lipschitz graph domains with controlled total surface area. This study is motivated by the following result proved by Peter Jones as a piece of his proof of the Analyst's Traveling Salesman Theorem in the complex plane: Any simply connected domain with finite boundary length can be decomposed into Lipschitz graph domains with total boundary length at most for some independent of . In this paper, we prove an analogous Lipschitz decomposition result in higher dimensions for domains with Reifenberg flat boundaries satisfying a uniform beta‐squared sum bound. We use similar techniques to show that domains with general Reifenberg flat or uniformly rectifiable boundaries admit similar Lipschitz decompositions while allowing the constituent domains to have bounded overlaps rather than be&nbsp;disjoint. https://escholarship.org/uc/item/5pg5p7j7 Mon, 28 Apr 2025 00:00:00 +0000 Krandel, Jared https://escholarship.org/uc/item/11g508gv “Paintings fade like flowers”: van Gogh’s prediction on the impact of age on paintings came true for most of his paintings. We have studied the consequences of this aging on the Sunflowers in a vase with a yellow background series, namely its original, F454, currently in London, and two replicates, F457, in Tokyo, and F458, in Amsterdam, which van Gogh painted using the original as a model. The background and flower renditions in those paintings have faded and turned brown, making them less vibrant that van Gogh had most likely intended. We have attempted to restore van Gogh’s intent using a computational approach based on data science. After identifications of regions of interest (ROI) within the three paintings F454, F457, and F458 that capture the flowers, stems of the flowers, and background, respectively, we studied the geometry of the color space (in RGB representation) occupied by those ROIs. By comparing those color spaces with those occupied by similar ROIs in photographs... https://escholarship.org/uc/item/11g508gv Thu, 10 Apr 2025 00:00:00 +0000 Liao, Shuting Koehl, Patrice Schultens, Jennifer https://orcid.org/0000-0003-3898-0051 Hsieh, Fushing https://orcid.org/0000-0002-9292-6980 https://escholarship.org/uc/item/50h793r6 Continued fractions and lines across the Stern–Brocot diagram https://escholarship.org/uc/item/50h793r6 Wed, 9 Apr 2025 00:00:00 +0000 Abramson, Heather Chesebro, Eric Cummins, Vivian Emlen, Cory Grady, Ryan Ke, Kenton https://escholarship.org/uc/item/4js6x7qf Farey recursive functions https://escholarship.org/uc/item/4js6x7qf Wed, 9 Apr 2025 00:00:00 +0000 Chesebro, Eric Emlen, Cory Ke, Kenton LaFontaine, Denise McKinnie, Kelly Rigby, Catherine https://escholarship.org/uc/item/8hn3k54s We study the lattice of submonoids of the uniform block permutation monoid containing the symmetric group (which is its group of units). We prove that this lattice is distributive under union and intersection by relating the submonoids containing the symmetric group to downsets in a new partial order on integer partitions. Furthermore, we show that the sizes of the J$$\mathscr {J}$$-classes of the uniform block permutation monoid are sums of squares of dimensions of irreducible modules of the monoid algebra. https://escholarship.org/uc/item/8hn3k54s Wed, 26 Mar 2025 00:00:00 +0000 Orellana, Rosa Saliola, Franco Schilling, Anne https://orcid.org/0000-0002-2601-7340 Zabrocki, Mike https://escholarship.org/uc/item/55v4v3jk Abstract: We introduce the immersion poset $$({\mathcal {P}}(n), \leqslant _I)$$ ( P ( n ) , ⩽ I ) on partitions, defined by $$\lambda \leqslant _I \mu $$ λ ⩽ I μ if and only if $$s_\mu (x_1, \ldots , x_N) - s_\lambda (x_1, \ldots , x_N)$$ ... https://escholarship.org/uc/item/55v4v3jk Wed, 12 Mar 2025 00:00:00 +0000 Johnston, Lisa Kenepp, David Nguyen, Evuilynn Paul, Digjoy Schilling, Anne https://orcid.org/0000-0002-2601-7340 Simone, Mary Claire Zhou, Regina https://escholarship.org/uc/item/3z3793p9 We give a corrigendum to our paper&nbsp;[4] entitled “A Demazure crystal construction for Schubert polynomials”. https://escholarship.org/uc/item/3z3793p9 Wed, 26 Feb 2025 00:00:00 +0000 Assaf, Sami Schilling, Anne https://orcid.org/0000-0002-2601-7340 https://escholarship.org/uc/item/7rd3q8kt The strong Haken theorem via sphere complexes https://escholarship.org/uc/item/7rd3q8kt Wed, 29 Jan 2025 00:00:00 +0000 Hensel, Sebastian Schultens, Jennifer https://escholarship.org/uc/item/7hf1p79d R-loops are a class of non-canonical nucleic acid structures that typically form during transcription when the nascent RNA hybridizes the DNA template strand, leaving the non-template DNA strand unpaired. These structures are abundant in nature and play important physiological and pathological roles. Recent research shows that DNA sequence and topology affect R-loops, yet it remains unclear how these and other factors contribute to R-loop formation. In this work, we investigate the link between nascent RNA folding and the formation of R-loops. We introduce tree-polynomials, a new class of representations of RNA secondary structures. A tree-polynomial representation consists of a rooted tree associated with an RNA secondary structure together with a polynomial that is uniquely identified with the rooted tree. Tree-polynomials enable accurate, interpretable and efficient data analysis of RNA secondary structures without pseudoknots. We develop a computational pipeline for investigating... https://escholarship.org/uc/item/7hf1p79d Mon, 20 Jan 2025 00:00:00 +0000 Liu, Pengyu Lusk, Jacob Jonoska, Nataša Vázquez, Mariel https://escholarship.org/uc/item/83s1w0m9 We study the structure of the set of all possible affine hyperplane sections of a convex polytope. We present two different cell decompositions of this set, induced by hyperplane arrangements. Using our decomposition, we bound the number of possible combinatorial types of sections and craft algorithms that compute optimal sections of the polytope according to various combinatorial and metric criteria, including sections that maximize the number of k-dimensional faces, maximize the volume, and maximize the integral of a polynomial. Our optimization algorithms run in polynomial time in fixed dimension, but the same problems show computational complexity hardness otherwise. Our tools can be extended to intersection with halfspaces and projections onto hyperplanes. Finally, we present several experiments illustrating our theorems and algorithms on famous polytopes. https://escholarship.org/uc/item/83s1w0m9 Fri, 17 Jan 2025 00:00:00 +0000 Brandenburg, Marie-charlotte De Loera, Jesus A https://orcid.org/0000-0002-9556-1112 Meroni, Chiara https://escholarship.org/uc/item/7hv0d6r8 We generalize R. P. Stanley's celebrated theorem that the h⁎-polynomial of the Ehrhart series of a rational polytope has nonnegative coefficients and is monotone under containment of polytopes. We show that these results continue to hold for weighted Ehrhart series where lattice points are counted with polynomial weights, as long as the weights are homogeneous polynomials decomposable as sums of products of linear forms that are nonnegative on the polytope. We also show nonnegativity of the h⁎-polynomial as a real-valued function for a larger family of weights. We explore the case when the weight function is the square of a single (arbitrary) linear form. We show stronger results for two-dimensional convex lattice polygons and give concrete examples showing tightness of the hypotheses. As an application, we construct a counterexample to a conjecture by Berg, Jochemko, and Silverstein on Ehrhart tensor polynomials. https://escholarship.org/uc/item/7hv0d6r8 Fri, 17 Jan 2025 00:00:00 +0000 Bajo, Esme Davis, Robert De Loera, Jesús A https://orcid.org/0000-0002-9556-1112 Garber, Alexey Mora, Sofía Garzón Jochemko, Katharina Yu, Josephine https://escholarship.org/uc/item/58b3n6qb We discuss five fundamental results of discrete mathematics: the lemmas of Sperner and Tucker from combinatorial topology and the theorems of Carathéodory, Helly, and Tverberg from combinatorial geometry. We explore their connections and emphasize their broad impact in application areas such as data science, game theory, graph theory, mathematical optimization, computational geometry, etc. https://escholarship.org/uc/item/58b3n6qb Fri, 17 Jan 2025 00:00:00 +0000 De Loera, Jesús A https://orcid.org/0000-0002-9556-1112 Goaoc, Xavier Meunier, Frédéric Mustafa, Nabil H https://escholarship.org/uc/item/5m15t3g3 Abstract: The classical Klein–Maskit combination theorems provide sufficient conditions to construct new Kleinian groups using old ones. There are two distinct but closely related combination theorems: the first deals with amalgamated free products, whereas the second deals with HNN extensions. This article gives analogs of both combination theorems for Anosov subgroups. https://escholarship.org/uc/item/5m15t3g3 Wed, 20 Nov 2024 00:00:00 +0000 Dey, Subhadip Kapovich, Michael https://orcid.org/0000-0001-5475-8755 https://escholarship.org/uc/item/1b48z46c It is an important problem in algebraic combinatorics to deduce the Schur function expansion of a symmetric function whose expansion in terms of the fundamental quasisymmetric function is known. For example, formulas are known for the fundamental expansion of a Macdonald symmetric function and for the plethysm of two Schur functions, while the Schur expansions of these expressions are still elusive. Based on work of Egge, Loehr and Warrington, Garsia and Remmel provided a method to obtain the Schur expansion from the fundamental expansion by replacing each quasisymmetric function by a Schur function (not necessarily indexed by a partition) and using straightening rules to obtain the Schur expansion. Here we provide a new method that only involves the coefficients of the quasisymmetric functions indexed by partitions and the quasi-Kostka matrix. As an application, we identify the lexicographically largest term in the Schur expansion of the plethysm of two Schur functions. We provide... https://escholarship.org/uc/item/1b48z46c Wed, 20 Nov 2024 00:00:00 +0000 Orellana, Rosa Saliola, Franco Schilling, Anne https://orcid.org/0000-0002-2601-7340 Zabrocki, Mike https://escholarship.org/uc/item/67z9r23d As machine learning powered decision-making becomes increasingly important in our daily lives, it is imperative to strive for fairness in the underlying data processing. We propose a pre-processing algorithm for fair data representation via which -objective supervised learning results in estimations of the Pareto frontier between prediction error and statistical disparity. Particularly, the present work applies the optimal affine transport to approach the post-processing Wasserstein barycenter characterization of the optimal fair -objective supervised learning via a pre-processing data deformation. Furthermore, we show that the Wasserstein geodesics from learning outcome marginals to their barycenter characterizes the Pareto frontier between -loss and total Wasserstein distance among the marginals. Numerical simulations underscore the advantages: (1) the pre-processing step is compositive with arbitrary conditional expectation estimation supervised learning methods and unseen... https://escholarship.org/uc/item/67z9r23d Wed, 13 Nov 2024 00:00:00 +0000 Xu, Shizhou Strohmer, Thomas https://escholarship.org/uc/item/2br289rp Complete characterizations of hyperbolic Coxeter groups with Sierpiński curve boundary and with Menger curve boundary https://escholarship.org/uc/item/2br289rp Wed, 23 Oct 2024 00:00:00 +0000 Danielski, Daniel Kapovich, Michael https://orcid.org/0000-0001-5475-8755 Świątkowski, Jacek https://escholarship.org/uc/item/4sn2p0kg Cardiac function is tightly regulated by the autonomic nervous system (ANS). Activation of the sympathetic nervous system increases cardiac output by increasing heart rate and stroke volume, while parasympathetic nerve stimulation instantly slows heart rate. Importantly, imbalance in autonomic control of the heart has been implicated in the development of arrhythmias and heart failure. Understanding of the mechanisms and effects of autonomic stimulation is a major challenge because synapses in different regions of the heart result in multiple changes to heart function. For example, nerve synapses on the sinoatrial node (SAN) impact pacemaking, while synapses on contractile cells alter contraction and arrhythmia vulnerability. Here, we present a multiscale neurocardiac modelling and simulator tool that predicts the effect of efferent stimulation of the sympathetic and parasympathetic branches of the ANS on the cardiac SAN and ventricular myocardium. The model includes a layered... https://escholarship.org/uc/item/4sn2p0kg Mon, 16 Sep 2024 00:00:00 +0000 Yang, Pei‐Chi https://orcid.org/0000-0002-5753-1131 Rose, Adam DeMarco, Kevin R Dawson, John RD Han, Yanxiao Jeng, Mao‐Tsuen Harvey, Robert D Santana, L Fernando Ripplinger, Crystal M Vorobyov, Igor https://orcid.org/0000-0002-4767-5297 Lewis, Timothy J Clancy, Colleen E https://escholarship.org/uc/item/8cw787s3 We prove nonemptyness of domains of proper discontinuity of Anosov groups of affine Lorentzian transformations of Rn. https://escholarship.org/uc/item/8cw787s3 Wed, 14 Aug 2024 00:00:00 +0000 Kapovich, Michael https://orcid.org/0000-0001-5475-8755 Leeb, Bernhard https://escholarship.org/uc/item/56j6m623 We prove a generalization of the classical Klein–Maskit combination theorem, in the free product case, in the setting of Anosov subgroups. Namely, if Γ A and Γ B are Anosov subgroups of a semisimple Lie group G of noncompact type, then under suitable topological assumptions, the group generated by Γ A and Γ B in G is again Anosov, and is naturally isomorphic to the free product Γ A∗ Γ B . Such a generalization was conjectured in Dey et al. (Math Z 293(1–2):551–578, 2019). https://escholarship.org/uc/item/56j6m623 Wed, 14 Aug 2024 00:00:00 +0000 Dey, Subhadip Kapovich, Michael https://orcid.org/0000-0001-5475-8755 https://escholarship.org/uc/item/57r3c1m9 Abstract: We compute the Borel–Moore homology of unramified affine Springer fibers for $\textrm{Gr}_n$ under the assumption that they are equivariantly formal and relate them to certain ideals discussed by Haiman. For $n=3$, we give an explicit description of these ideals, compute their Hilbert series, generators, and relations, and compare them to generalized $(q,t)$-Catalan numbers. We also compare the homology to the Khovanov–Rozansky homology of the associated link, and prove a version of a conjecture of Oblomkov, Rasmussen, and Shende in this case. https://escholarship.org/uc/item/57r3c1m9 Thu, 25 Jul 2024 00:00:00 +0000 Turner, Joshua P https://escholarship.org/uc/item/3415f5nh Many microorganisms propel themselves through complex media by deforming their flagella. The beat is thought to emerge from interactions between forces of the surrounding fluid, the passive elastic response from deformations of the flagellum and active forces from internal molecular motors. The beat varies in response to changes in the fluid rheology, including elasticity, but there are limited data on how systematic changes in elasticity alter the beat. This work analyses a related problem with fixed-strength driving force: the emergence of beating of an elastic planar filament driven by a follower force at the tip of a viscoelastic fluid. This analysis examines how the onset of oscillations depends on the strength of the force and viscoelastic parameters. Compared to a Newtonian fluid, it takes more force to induce the instability in viscoelastic fluids, and the frequency of the oscillation is higher. The linear analysis predicts that the frequency increases with the fluid relaxation... https://escholarship.org/uc/item/3415f5nh Tue, 25 Jun 2024 00:00:00 +0000 Link, Kathryn G Guy, Robert D https://orcid.org/0000-0002-0602-7015 Thomases, Becca Arratia, Paulo E https://escholarship.org/uc/item/2f707966 It is widely recognized that the three-dimensional (3D) architecture of eukaryotic chromatin plays an important role in processes such as gene regulation and cancer-driving gene fusions. Observing or inferring this 3D structure at even modest resolutions had been problematic, since genomes are highly condensed and traditional assays are coarse. However, recently devised high-throughput molecular techniques have changed this situation. Notably, the development of a suite of chromatin conformation capture (CCC) assays has enabled elicitation of contacts-spatially close chromosomal loci-which have provided insights into chromatin architecture. Most analysis of CCC data has focused on the contact level, with less effort directed toward obtaining 3D reconstructions and evaluating the accuracy and reproducibility thereof. While questions of accuracy must be addressed experimentally, questions of reproducibility can be addressed statistically-the purpose of this paper. We use a constrained... https://escholarship.org/uc/item/2f707966 Mon, 17 Jun 2024 00:00:00 +0000 Segal, Mark R Xiong, Hao Capurso, Daniel Vazquez, Mariel https://orcid.org/0000-0001-8328-6806 Arsuaga, Javier https://escholarship.org/uc/item/6k09x9vf Differential privacy is a mathematical concept that provides an information-theoretic security guarantee. While differential privacy has emerged as a de facto standard for guaranteeing privacy in data sharing, the known mechanisms to achieve it come with some serious limitations. Utility guarantees are usually provided only for a fixed, a priori specified set of queries. Moreover, there are no utility guarantees for more complex—but very common—machine learning tasks such as clustering or classification. In this paper we overcome some of these limitations. Working with metric privacy, a powerful generalization of differential privacy, we develop a polynomial-time algorithm that creates a private measure from a data set. This private measure allows us to efficiently construct private synthetic data that are accurate for a wide range of statistical analysis tools. Moreover, we prove an asymptotically sharp min-max result for private measures and synthetic data in general compact... https://escholarship.org/uc/item/6k09x9vf Wed, 12 Jun 2024 00:00:00 +0000 Boedihardjo, March Strohmer, Thomas Vershynin, Roman https://escholarship.org/uc/item/3rm3m38c Mexican, Puerto Rican, and Central American Ancestry (MPRCA) individuals represent 82% of US Latinos. An intergenerational group of MPRCA women and allies met to discuss persistent underrepresentation of MPRCA women in STEM, identifying multi-level challenges and solutions. Implementation of these solutions is important and will benefit MPRCA women and the entire academic community. https://escholarship.org/uc/item/3rm3m38c Thu, 9 May 2024 00:00:00 +0000 Werner Washburne, Maggie Trejo, JoAnn https://orcid.org/0000-0003-4405-6228 Zambrana, Ruth Enid Zavala, Maria Elena Martinic, Alice Riestra, Angelica Delgado, Tracie Edwards, Staci Escobar, Thelma Jamison-McClung, Denneal Vazquez, Mariel https://orcid.org/0000-0001-8328-6806 Vera, Iset Guerra, Michelle Marinez, Diana I Gonzalez, Elma Rodriguez, Raymond L https://orcid.org/0000-0002-6105-2571 https://escholarship.org/uc/item/1vw4k87g Our previous multiscale graph basis dictionaries/graph signal transforms—Generalized Haar-Walsh Transform (GHWT); Hierarchical Graph Laplacian Eigen Transform (HGLET); Natural Graph Wavelet Packets (NGWPs); and their relatives—were developed for analyzing data recorded on vertices of a given graph. In this article, we propose their generalization for analyzing data recorded on edges, faces (i.e., triangles), or more generally κ$$\kappa $$-dimensional simplices of a simplicial complex (e.g., a triangle mesh of a manifold). The key idea is to use the Hodge Laplacians and their variants for hierarchical partitioning of a set of κ$$\kappa $$-dimensional simplices in a given simplicial complex, and then build localized basis functions on these partitioned subsets. We demonstrate their usefulness for data representation on both illustrative synthetic examples and real-world simplicial complexes generated from a co-authorship/citation dataset and an ocean current/flow dataset. https://escholarship.org/uc/item/1vw4k87g Wed, 8 May 2024 00:00:00 +0000 Saito, Naoki https://orcid.org/0000-0001-5234-4719 Schonsheck, Stefan C Shvarts, Eugene https://escholarship.org/uc/item/8bz0f64n Computational complexity and 3–manifolds and zombies https://escholarship.org/uc/item/8bz0f64n Wed, 24 Apr 2024 00:00:00 +0000 Kuperberg, Greg Samperton, Eric https://escholarship.org/uc/item/70d4c5bf The Cartan–Hadamard conjecture and the Little Prince https://escholarship.org/uc/item/70d4c5bf Wed, 24 Apr 2024 00:00:00 +0000 Kloeckner, Benoît R Kuperberg, Greg https://escholarship.org/uc/item/66h3b8md Identifying lens spaces in polynomial time https://escholarship.org/uc/item/66h3b8md Wed, 24 Apr 2024 00:00:00 +0000 Kuperberg, Greg https://escholarship.org/uc/item/5js4159f Coloring invariants of knots and links are often intractable https://escholarship.org/uc/item/5js4159f Wed, 24 Apr 2024 00:00:00 +0000 Kuperberg, Greg Samperton, Eric https://escholarship.org/uc/item/090080wv Abstract: Let $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\Gamma $ be a compact tropical curve (or metric graph) of genus&nbsp;$g$. Using the theory of tropical theta functions, Mikhalkin and Zharkov proved that there is a canonical effective representative (called a break divisor) for each linear equivalence class of divisors of degree $g$ on&nbsp;$\Gamma $. We present a new combinatorial proof of the fact that there is a unique break divisor in each equivalence class, establishing in the process an ‘integral’ version of this result which is of independent interest. As an application, we provide a ‘geometric proof’ of (a dual version of) Kirchhoff’s celebrated matrix–tree theorem. Indeed, we show that each weighted graph model $G$ for $\Gamma $ gives rise to a canonical polyhedral decomposition of the $g$-dimensional... https://escholarship.org/uc/item/090080wv Wed, 24 Apr 2024 00:00:00 +0000 AN, YANG BAKER, MATTHEW KUPERBERG, GREG SHOKRIEH, FARBOD https://escholarship.org/uc/item/7hg208q7 The allometric trophic network (ATN) framework for modeling population dynamics has provided numerous insights into ecosystem functioning in recent years. Herein we extend ATN modeling of the intertidal ecosystem off central Chile to include empirical data on pelagic chlorophyll-a concentration. This intertidal community requires subsidy of primary productivity to support its rich ecosystem. Previous work models this subsidy using a constant rate of phytoplankton input to the system. However, data shows pelagic subsidies exhibit highly variable, pulse-like behavior. The primary contribution of our work is incorporating this variable input into ATN modeling to simulate how this ecosystem may respond to pulses of pelagic phytoplankton. Our model results show that: (1) closely related sea snails respond differently to phytoplankton variability, which is explained by the underlying network structure of the food web; (2) increasing the rate of pelagic-intertidal mixing increases fluctuations... https://escholarship.org/uc/item/7hg208q7 Mon, 1 Apr 2024 00:00:00 +0000 Duckwall, Casey Largier, John L https://orcid.org/0000-0003-4273-4443 Wieters, Evie A Valdovinos, Fernanda S https://escholarship.org/uc/item/96v2r876 In this article, we evaluate the challenges and best practices associated with the Markov bases approach to sampling from conditional distributions. We provide insights and clarifications after 25 years of the publication of the Fundamental theorem for Markov bases by Diaconis and Sturmfels. In addition to a literature review, we prove three new results on the complexity of Markov bases in hierarchical models, relaxations of the fibers in log-linear models, and limitations of partial sets of moves in providing an irreducible Markov chain. Supplementary materials for this article are available online. https://escholarship.org/uc/item/96v2r876 Wed, 27 Mar 2024 00:00:00 +0000 Almendra-Hernández, Félix De Loera, Jesús A https://orcid.org/0000-0002-9556-1112 Petrović, Sonja https://escholarship.org/uc/item/92p1f9k1 Motivated by the analysis of the performance of the simplex method, we study the behavior of families of pivot rules of linear programs. We introduce normalized-weight pivot rules which are fundamental for the following reasons: First, they are memory-less, in the sense that the pivots are governed by local information encoded by an arborescence. Second, many of the most used pivot rules belong to that class, and we show this subclass is critical for understanding the complexity of all pivot rules. Finally, normalized-weight pivot rules can be parametrized in a natural continuous manner. The latter leads to the existence of two polytopes, the pivot rule polytopes and the neighbotopes, that capture the behavior of normalized-weight pivot rules on polytopes and linear programs. We explain their face structure in terms of multi-arborescences and compute upper bounds on the number of coherent arborescences, that is, vertices of our polytopes. We introduce a normalized-weight pivot... https://escholarship.org/uc/item/92p1f9k1 Wed, 27 Mar 2024 00:00:00 +0000 Black, Alexander E De Loera, Jesús A https://orcid.org/0000-0002-9556-1112 Lütjeharms, Niklas Sanyal, Raman https://escholarship.org/uc/item/75c5j84r In this paper, we explore affine semigroup versions of the convex geometry theorems of Helly, Tverberg, and Carathéodory. Additionally, we develop a new theory of colored affine semigroups, where the semigroup generators each receive a color and the elements of the semigroup take into account the colors used (the classical theory of affine semigroups coincides with the case in which all generators have the same color). We prove an analog of Tverberg’s theorem and colorful Helly’s theorem for semigroups, as well as a version of colorful Carathéodory’s theorem for cones. We also demonstrate that colored numerical semigroups are particularly rich by introducing a colored version of the Frobenius number. https://escholarship.org/uc/item/75c5j84r Wed, 27 Mar 2024 00:00:00 +0000 De Loera, Jesús A https://orcid.org/0000-0002-9556-1112 O’Neill, Christopher Wang, Chengyang https://escholarship.org/uc/item/6p3161d7 We show that, under a natural probability distribution, random monomial ideals will almost always have minimal free resolutions of maximal length; that is, the projective dimension will almost always be n, where n is the number of variables in the polynomial ring. As a consequence we prove that Cohen.Macaulayness is a rare property. We characterize when a random monomial ideal is generic/strongly generic, and when it is Scarf.i.e., when the algebraic Scarf complex of M S = k[x1, . . . , xn] gives a minimal free resolution of S/M. It turns out, outside of a very specific ratio of model parameters, random monomial ideals are Scarf only when they are generic. We end with a discussion of the average magnitude of Betti numbers. https://escholarship.org/uc/item/6p3161d7 Wed, 27 Mar 2024 00:00:00 +0000 De Loera, Jesús A https://orcid.org/0000-0002-9556-1112 Hoşten, Serkan Krone, Robert Silverstein, Lily https://escholarship.org/uc/item/6mk1d53m Motivated by the problem of bounding the number of iterations of the simplex algorithm, we investigate the possible lengths of monotone paths followed inside the oriented graphs of polyhedra (oriented by the objective function). We consider both the shortest and the longest monotone paths and estimate the monotone diameter and height of polyhedra. Our analysis applies to transportation polytopes, matroid polytopes, matching polytopes, shortest-path polytopes, and the traveling salesman polytope, among others. We begin by showing that combinatorial cubes have monotone diameter and Bland simplex height upper bounded by their dimension and that in fact all monotone paths of zonotopes are no larger than the number of edge directions of the zonotope. We later use this to show that several polytopes have polynomial-size monotone diameter. In contrast, we show that for many well-known combinatorial polytopes, the height is at least exponential. Surprisingly, for some famous pivot rules,... https://escholarship.org/uc/item/6mk1d53m Wed, 27 Mar 2024 00:00:00 +0000 Blanchard, Moïse De Loera, Jesús A https://orcid.org/0000-0002-9556-1112 Louveaux, Quentin https://escholarship.org/uc/item/6b25c1h0 Inspired by the study of random graphs and simplicial complexes, and motivated by the need to understand average behavior of ideals, we propose and study probabilistic models of random monomial ideals. We prove theorems about the probability distributions, expectations and thresholds for events involving monomial ideals with given Hilbert function, Krull dimension, and first graded Betti numbers, and show that our models generalize several well-known random simplicial complex models. Finally we present several experimentally-backed conjectures about regularity, projective dimension, strong genericity, and Cohen–Macaulayness of random monomial ideals. https://escholarship.org/uc/item/6b25c1h0 Wed, 27 Mar 2024 00:00:00 +0000 De Loera, Jesús A https://orcid.org/0000-0002-9556-1112 Petrović, Sonja Silverstein, Lily Stasi, Despina Wilburne, Dane https://escholarship.org/uc/item/5kt2c509 Machine learning algorithms typically rely on optimization subroutines and are well known to provide very effective outcomes for many types of problems. Here, we flip the reliance and ask the reverse question: can machine learning algorithms lead to more effective outcomes for optimization problems? Our goal is to train machine learning methods to automatically improve the performance of optimization and signal processing algorithms. As a proof of concept, we use our approach to improve two popular data processing subroutines in data science: stochastic gradient descent and greedy methods in compressed sensing. We provide experimental results that demonstrate the answer is “yes”, machine learning algorithms do lead to more effective outcomes for optimization problems, and show the future potential for this research direction. In addition to our experimental work, we prove relevant Probably Approximately Correct (PAC) learning theorems for our problems of interest. More precisely,... https://escholarship.org/uc/item/5kt2c509 Wed, 27 Mar 2024 00:00:00 +0000 De Loera, Jesús A https://orcid.org/0000-0002-9556-1112 Haddock, Jamie Ma, Anna Needell, Deanna https://escholarship.org/uc/item/5cz296g3 In the early 1990s, Billera and Sturmfels introduced the monotone path polytope (MPP), an important case of the general theory of fiber polytopes, which has led to remarkable combinatorics. Given a pair (P, φ) of a polytope P and a linear functional φ , the MPP is obtained from averaging the fibers of the projection φ(P) . They also showed that MPPs of (regular) simplices and hyper-cubes are combinatorial cubes and permutahedra respectively. As a natural follow-up we investigate the monotone paths of cross-polytopes for a generic linear functional φ . We show the face lattice of the MPP of the cross-polytope is isomorphic to the lattice of intervals in the sign poset from oriented matroid theory. We also describe its f-vector, some geometric realizations, an irredundant inequality description, the 1-skeleton and we compute its diameter. In contrast to the case of simplices and hyper-cubes, monotone paths on cross-polytopes are not always coherent. https://escholarship.org/uc/item/5cz296g3 Wed, 27 Mar 2024 00:00:00 +0000 Black, Alexander E De Loera, Jesús A https://orcid.org/0000-0002-9556-1112 https://escholarship.org/uc/item/5b6972vn Tverberg’s theorem says that a set with sufficiently many points in Rd can always be partitioned into m parts so that the (m- 1) -simplex is the (nerve) intersection pattern of the convex hulls of the parts. The main results of our paper demonstrate that Tverberg’s theorem is just a special case of a more general situation, where other simplicial complexes must always arise as nerve complexes, as soon as the number of points is large enough. We prove that, given a set with sufficiently many points, all trees and all cycles can also be induced by at least one partition of the point set. We also discuss how some simplicial complexes can never be achieved this way, even for arbitrarily large sets of points. https://escholarship.org/uc/item/5b6972vn Wed, 27 Mar 2024 00:00:00 +0000 De Loera, Jesús A https://orcid.org/0000-0002-9556-1112 Hogan, Thomas A Oliveros, Deborah Yang, Dominic https://escholarship.org/uc/item/20f776vg We study bases of the lattice generated by the cycles of an undirected graph, defined as the integer linear combinations of the 0/1-incidence vectors of cycles. We prove structural results for this lattice, including explicit formulas for its dimension and determinant, and we present efficient algorithms to construct lattice bases, using only cycles as generators, in quadratic time. By algebraic considerations, we relate these results to the more general setting with coefficients from an arbitrary Abelian group. Our results generalize classical results for the vector space of cycles of a graph over the binary field to the case of an arbitrary field. https://escholarship.org/uc/item/20f776vg Wed, 27 Mar 2024 00:00:00 +0000 Averkov, G Chavez, A De Loera, JA https://orcid.org/0000-0002-9556-1112 Gillespie, B https://escholarship.org/uc/item/1c97d268 The complexity of Philip Wolfe's method for the minimum Euclidean-norm point problem over a convex polytope has remained unknown since he proposed the method in 1974. The method is important because it is used as a subroutine for one of the most practical algorithms for submodular function minimization. We present the first example that Wolfe's method takes exponential time. Additionally, we improve previous results to show that linear programming reduces in strongly polynomial time to the minimum norm point problem over a simplex. https://escholarship.org/uc/item/1c97d268 Wed, 27 Mar 2024 00:00:00 +0000 De Loera, Jesús A https://orcid.org/0000-0002-9556-1112 Haddock, Jamie Rademacher, Luis