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Kevin Gomez

Publications and source records attributed to Kevin Gomez.

13 recordsLinked to original sources
arXiv2025

Zeros of Hecke polynomials arising from weak eigenforms

We attach Hecke polynomials $P_n(F;x)$ to weak Hecke eigenforms $F$ of weight $2-k$ and show that, for large $n$, every zero is simple and lies in $[0,1728]$. The construction pulls back a weakly holomorphic Hecke combination of $F$ along $j$; the analysis follows Hecke orbits on the unit-circle arc $\mathcal{A}$, isolating a dominant "cosine" term and controlling the tail via Maass-Poincar\'e series and Whittaker/Bessel bounds. This extends the Rankin--Swinnerton-Dyer/Asai--Kaneko--Ninomiya picture from holomorphic forms to a broad class of harmonic Maass forms and yields a clean degree-monicity formula and simple criteria for zeros at $0$ and $1728$.

math.NT↗
arXiv2025

Hecke polynomials for the mock modular form arising from the Delta-function

We consider a mock modular form $M_{\Delta}(\tau)$ that arises naturally from Ramanujan's Delta-function. It is a weight $-10$ harmonic Maass form whose nonholomorphic part is the "period integral function'' of $\Delta(\tau)$. The Hecke operator $T_{-10}(m)$ acts on this mock modular form in terms of Ramanujan's $\tau(m)$ and a monic degree $m$ polynomial $F_m(x),$ evaluated at $x=j(\tau).$ In analogy with results by Asai, Kaneko, and Ninomiya on the zeros of Hecke polynomials for the $j$-function, we prove that the zeros of each $F_m(x)$, including $x=0$ and $x=1728,$ are distinct and lie in $[0, 1728]$. Additionally, as $m \to +\infty,$ these zeros become equidistributed in $[0, 1728].$

math.NT↗
arXiv2024

Pentagonal number recurrence relations for $p(n)$

We revisit Euler's partition function recurrence, which asserts, for integers $n\geq 1,$ that $$ p(n)=p(n-1)+p(n-2)-p(n-5)-p(n-7)+\dots = \sum_{k\in \mathbb{Z}\setminus \{0\}} (-1)^{k+1} p(n-\omega(k)), $$ where $\omega(m):=(3m^2+m)/2$ is the $m$th pentagonal number. We prove that this classical result is the $\nu=0$ case of an infinite family of ``pentagonal number'' recurrences. For each $\nu\geq 0,$ we prove for positive $n$ that $$ p(n)=\frac{1}{g_{\nu}(n,0)}\left(\alpha_{\nu}\cdot \sigma_{2\nu-1}(n)+ \mathrm{Tr}_{2\nu}(n) +\sum_{k\in \mathbb{Z}\setminus \{0\}} (-1)^{k+1} g_{\nu}(n,k)\cdot p(n-\omega(k))\right), $$ where $\sigma_{2\nu-1}(n)$ is a divisor function, $\mathrm{Tr}_{2\nu}(n)$ is the $n$th weight $2\nu$ Hecke trace of values of special twisted quadratic Dirichlet series, and each $g_{\nu}(n,k)$ is a polynomial in $n$ and $k.$ The $\nu=6$ case can be viewed as a partition theoretic formula for Ramanujan's tau-function, as we have $$ \mathrm{Tr}_{12}(n)=-\frac{33108590592}{691}\cdot \tau(n). $$

math.NT↗
arXiv2024

MacMahonesque partition functions detect sets related to primes

Recent work by Craig, van Ittersum, and Ono constructs explicit expressions in the partition functions of MacMahon that detect the prime numbers. Furthermore, they define generalizations, the MacMahonesque functions, and prove there are infinitely many such expressions in these functions. Here, we show how to modify and adapt their construction to detect cubes of primes as well as primes in arithmetic progressions.

math.NT↗
arXiv2024

CHIME: Energy-Efficient STT-RAM-based Concurrent Hierarchical In-Memory Processing

Processing-in-cache (PiC) and Processing-in-memory (PiM) architectures, especially those utilizing bit-line computing, offer promising solutions to mitigate data movement bottlenecks within the memory hierarchy. While previous studies have explored the integration of compute units within individual memory levels, the complexity and potential overheads associated with these designs have often limited their capabilities. This paper introduces a novel PiC/PiM architecture, Concurrent Hierarchical In-Memory Processing (CHIME), which strategically incorporates heterogeneous compute units across multiple levels of the memory hierarchy. This design targets the efficient execution of diverse, domain-specific workloads by placing computations closest to the data where it optimizes performance, energy consumption, data movement costs, and area. CHIME employs STT-RAM due to its various advantages in PiC/PiM computing, such as high density, low leakage, and better resiliency to data corruption from activating multiple word lines. We demonstrate that CHIME enhances concurrency and improves compute unit utilization at each level of the memory hierarchy. We present strategies for exploring the design space, grouping, and placing the compute units across the memory hierarchy. Experiments reveal that, compared to the state-of-the-art bit-line computing approaches, CHIME achieves significant speedup and energy savings of 57.95% and 78.23% for various domain-specific workloads, while reducing the overheads associated with single-level compute designs.

cs.CY↗
arXiv2022

The second shifted difference of partitions and its applications

A number of recent papers have estimated ratios of the partition function $p(n-j)/p(n)$, which appears in many applications. Here, we prove an easy-to-use effective bound on these ratios. Using this, we then study second shifted difference of partitions, $f(j,n):= p(n) -2p(n-j) +p(n-2j)$, and give another easy-to-use estimate of $f(j,n)$. As applications of these, we prove a shifted convexity property of $p(n)$, as well as giving new estimates of the $k$-rank partition function $N_k(m,n)$ and non-$k$-ary partitions along with their differences.

math.NT↗
arXiv2021

Infinite families of crank functions, Stanton-type conjectures, and unimodality

Dyson's rank function and the Andrews--Garvan crank function famously give combinatorial witnesses for Ramanujan's partition function congruences modulo 5, 7, and 11. While these functions can be used to show that the corresponding sets of partitions split into 5, 7, or 11 equally sized sets, one may ask how to make the resulting bijections between partitions organized by rank or crank combinatorially explicit. Stanton recently made conjectures which aim to uncover a deeper combinatorial structure along these lines, where it turns out that minor modifications of the rank and crank are required. Here, we prove two of these conjectures. We also provide abstract criteria for quotients of polynomials by certain cyclotomic polynomials to have non-negative coefficients based on unimodality and symmetry. Furthermore, we extend Stanton's conjecture to an infinite family of cranks. This suggests further applications to other combinatorial objects. We also discuss numerical evidence for our conjectures, connections with other analytic conjectures such as the distribution of partition ranks.

math.NT↗
arXiv2021

Supersingular Loci from Traces of Hecke Operators

A classical observation of Deligne shows that, for any prime $p \geq 5$, the divisor polynomial of the Eisenstein series $E_{p-1}(z)$ mod $p$ is closely related to the supersingular polynomial at $p$, $$S_p(x) := \prod_{E/\bar{\mathbb{F}}_p \text{ supersingular}}(x-j(E)) \in \mathbb{F}_p[x].$$ Deuring, Hasse, and Kaneko and Zagier found other families of modular forms which also give the supersingular polynomial at $p$. In a new approach, we prove an analogue of Deligne's result for the Hecke trace forms $T_k(z)$ defined by the Hecke action on the space of cusp forms $S_k$. We use the Eichler-Selberg trace formula to identify congruences between trace forms of different weights mod $p$, and then relate their divisor polynomials to $S_p(x)$ using Deligne's observation.

math.NT↗
arXiv2020

End-to-End Memristive HTM System for Pattern Recognition and Sequence Prediction

Neuromorphic systems that learn and predict from streaming inputs hold significant promise in pervasive edge computing and its applications. In this paper, a neuromorphic system that processes spatio-temporal information on the edge is proposed. Algorithmically, the system is based on hierarchical temporal memory that inherently offers online learning, resiliency, and fault tolerance. Architecturally, it is a full custom mixed-signal design with an underlying digital communication scheme and analog computational modules. Therefore, the proposed system features reconfigurability, real-time processing, low power consumption, and low-latency processing. The proposed architecture is benchmarked to predict on real-world streaming data. The network's mean absolute percentage error on the mixed-signal system is 1.129X lower compared to its baseline algorithm model. This reduction can be attributed to device non-idealities and probabilistic formation of synaptic connections. We demonstrate that the combined effect of Hebbian learning and network sparsity also plays a major role in extending the overall network lifespan. We also illustrate that the system offers 3.46X reduction in latency and 77.02X reduction in power consumption when compared to a custom CMOS digital design implemented at the same technology node. By employing specific low power techniques, such as clock gating, we observe 161.37X reduction in power consumption.

cs.ET↗
arXiv2019

Graph Sampling with Distributed In-Memory Dataflow Systems

Given a large graph, a graph sample determines a subgraph with similar characteristics for certain metrics of the original graph. The samples are much smaller thereby accelerating and simplifying the analysis and visualization of large graphs. We focus on the implementation of distributed graph sampling for Big Data frameworks and in-memory dataflow systems such as Apache Spark or Apache Flink. We evaluate the scalability of the new implementations and analyze to what degree the sampling approaches preserve certain graph metrics compared to the original graph. The latter analysis also uses comparative graph visualizations. The presented methods will be open source and be integrated into Gradoop, a system for distributed graph analytics.

cs.DC↗
arXiv2016

Non-volatile Hierarchical Temporal Memory: Hardware for Spatial Pooling

Hierarchical Temporal Memory (HTM) is a biomimetic machine learning algorithm imbibing the structural and algorithmic properties of the neocortex. Two main functional components of HTM that enable spatio-temporal processing are the spatial pooler and temporal memory. In this research, we explore a scalable hardware realization of the spatial pooler closely coupled with the mathematical formulation of spatial pooler. This class of neuromorphic algorithms are advantageous in solving a subset of the future engineering problems by extracting nonintuitive patterns in complex data. The proposed architecture, Non-volatile HTM (NVHTM), leverages large-scale solid state flash memory to realize a optimal memory organization, area and power envelope. A behavioral model of NVHTM is evaluated against the MNIST dataset, yielding 91.98% classification accuracy. A full custom layout is developed to validate the design in a TSMC 180nm process. The area and power profile of the spatial pooler are 30.538mm2 and 64.394mW, respectively. This design is a proof-of-concept that storage processing is a viable platform for large scale HTM network models.

cs.AR↗
arXiv2016

Predicting patterns of long-term adaptation and extinction with population genetics

Population genetics struggles to model extinction; standard models track the relative rather than absolute fitness of genotypes, while the exceptions describe only the short-term transition from imminent doom to evolutionary rescue. But extinction can result from failure to adapt not only to catastrophes, but also to a backlog of environmental challenges. We model long-term evolution to long series of small challenges, where fitter populations reach higher population sizes. The population's long-term fitness dynamic is well approximated by a simple stochastic Markov chain model. Long-term persistence occurs when the rate of adaptation exceeds the rate of environmental deterioration for some genotypes. Long-term persistence times are consistent with typical fossil species persistence times of several million years. Immediately preceding extinction, fitness declines rapidly, appearing as though a catastrophe disrupted a stably established population, even though gradual evolutionary processes are responsible. New populations go through an establishment phase where, despite being demographically viable, their extinction risk is elevated. Should the population survive long enough, extinction risk later becomes constant over time.

q-bio.PE↗