Arithmetic Counts Research Articles

Arithmetic counts of tropical plane curves and their properties

Abstract Recently, the first and third author proved a correspondence theorem which recovers the Levine- Welschinger invariants of toric del Pezzo surfaces as a count of tropical curves weighted with arithmetic multiplicities. In this paper, we study properties of the arithmetic count ofplane tropical curves satisfying point conditions. We prove that this count is independent of the configuration of point conditions. Moreover, a Caporaso- Harris formula for the arithmetic count of plane tropical curves is obtained by moving one point to the very left. Repeating this process until all point conditions are stretched, we obtain an enriched count of floor diagrams which coincides with the tropical count. Finally, we prove polynomiality properties for the arithmetic counts using floor diagrams.

EULER CLASSES: SIX-FUNCTORS FORMALISM, DUALITIES, INTEGRALITY AND LINEAR SUBSPACES OF COMPLETE INTERSECTIONS

AbstractWe equate various Euler classes of algebraic vector bundles, including those of [12] and one suggested by M. J. Hopkins, A. Raksit, and J.-P. Serre. We establish integrality results for this Euler class and give formulas for local indices at isolated zeros, both in terms of the six-functors formalism of coherent sheaves and as an explicit recipe in the commutative algebra of Scheja and Storch. As an application, we compute the Euler classes enriched in bilinear forms associated to arithmetic counts ofd-planes on complete intersections in$\mathbb P^n$in terms of topological Euler numbers over$\mathbb {R}$and$\mathbb {C}$.

An arithmetic count of the lines on a smooth cubic surface

We give an arithmetic count of the lines on a smooth cubic surface over an arbitrary field$k$, generalizing the counts that over${\mathbf {C}}$there are$27$lines, and over${\mathbf {R}}$the number of hyperbolic lines minus the number of elliptic lines is$3$. In general, the lines are defined over a field extension$L$and have an associated arithmetic type$\alpha$in$L^*/(L^*)^2$. There is an equality in the Grothendieck–Witt group$\operatorname {GW}(k)$of$k$,\[ \sum_{\text{lines}} \operatorname{Tr}_{L/k} \langle \alpha \rangle = 15 \cdot \langle 1 \rangle + 12 \cdot \langle -1 \rangle, \]where$\operatorname {Tr}_{L/k}$denotes the trace$\operatorname {GW}(L) \to \operatorname {GW}(k)$. Taking the rank and signature recovers the results over${\mathbf {C}}$and${\mathbf {R}}$. To do this, we develop an elementary theory of the Euler number in$\mathbf {A}^1$-homotopy theory for algebraic vector bundles. We expect that further arithmetic counts generalizing enumerative results in complex and real algebraic geometry can be obtained with similar methods.

An arithmetic count of the lines meeting four lines in 𝐏³

We enrich the classical count that there are two complex lines meeting four lines in space to an equality of isomorphism classes of bilinear forms. For any field k k , this enrichment counts the number of lines meeting four lines defined over k k in P k 3 \mathbf {P}^3_k , with such lines weighted by their fields of definition together with information about the cross-ratio of the intersection points and spanning planes. We generalize this example to an infinite family of such enrichments, obtained using an Euler number in A 1 \mathbf {A}^1 -homotopy theory. The classical counts are recovered by taking the rank of the bilinear forms.

An Arithmetic Count of the Lines on a Smooth Cubic Surface

An Arithmetic Count of the Lines on a Smooth Cubic Surface

Assessing food security and environmental protection in Mexico with a GIS-based Food Environmental Efficiency index

Trends in food security and environmental protection are usually reported separately and at national level, which may be a great limitation to the assessment of regional policies seeking to improve food self-sufficiency, reduce poverty, and at the same time conserve biodiversity. In this study, a spatially explicit, quantitative index relates national and regional trends of food security with trends of land use change in Mexico. Food security was estimated through aspects of food self-sufficiency (production and consumption patterns of basic staple crops and livestock) and food access (based on the marginalization level of households). Land use change was estimated from the official INEGI Land Use and Land Cover cartography. The Food Environmental Efficiency (FEE) Index was calculated for each ecoregion of Mexico over the past 40 years based on an arithmetic count of significant correlations between food security and land use change. Trends at national level suggest a continuous environmental degradation and no improvement in food security except for maize self-sufficiency. At ecoregion level, the FEE index indicates that livestock expansion in the three most affected ecoregions is associated with a decrease in food security and that extensive cropland expansion is associated with an increase in food security in only one of them. The FEE index proved useful for the assessment of land use policies, since it can weigh regional contributions to food security and environmental tendencies.

Instruction scheduling heuristic for an efficient FFT in VLIW processors with balanced resource usage

The fast Fourier transform (FFT) is perhaps today’s most ubiquitous algorithm used with digital data; hence, it is still being studied extensively. Besides the benefit of reducing the arithmetic count in the FFT algorithm, memory references and scheme’s projection on processor’s architecture are critical for a fast and efficient implementation. One of the main bottlenecks is in the long latency memory accesses to butterflies’ legs and in the redundant references to twiddle factors. In this paper, we describe a new FFT implementation on high-end very long instruction word (VLIW) digital signal processors (DSP), which presents improved performance in terms of clock cycles due to the resulting low-level resource balance and to the reduced memory accesses of twiddle factors. The method introduces a tradeoff parameter between accuracy and speed. Additionally, we suggest a cache-efficient implementation methodology for the FFT, dependently on the provided VLIW hardware resources and cache structure. Experimental results on a TI VLIW DSP show that our method reduces the number of clock cycles by an average of 51 % (2 times acceleration) when compared to the most assembly-optimized and vendor-tuned FFT libraries. The FFT was generated using an instruction-level scheduling heuristic. It is a modulo-based register-sensitive scheduling algorithm, which is able to compute an aggressively efficient sequence of VLIW instructions for the FFT, maximizing the parallelism rate and minimizing clock cycles and register usage.

Why Mental Arithmetic Counts: Brain Activation during Single Digit Arithmetic Predicts High School Math Scores

Do individual differences in the brain mechanisms for arithmetic underlie variability in high school mathematical competence? Using functional magnetic resonance imaging, we correlated brain responses to single digit calculation with standard scores on the Preliminary Scholastic Aptitude Test (PSAT) math subtest in high school seniors. PSAT math scores, while controlling for PSAT Critical Reading scores, correlated positively with calculation activation in the left supramarginal gyrus and bilateral anterior cingulate cortex, brain regions known to be engaged during arithmetic fact retrieval. At the same time, greater activation in the right intraparietal sulcus during calculation, a region established to be involved in numerical quantity processing, was related to lower PSAT math scores. These data reveal that the relative engagement of brain mechanisms associated with procedural versus memory-based calculation of single-digit arithmetic problems is related to high school level mathematical competence, highlighting the fundamental role that mental arithmetic fluency plays in the acquisition of higher-level mathematical competence.

Graph expansion and communication costs of fast matrix multiplication

The communication cost of algorithms (also known as I/O-complexity) is shown to be closely related to the expansion properties of the corresponding computation graphs. We demonstrate this on Strassen's and other fast matrix multiplication algorithms, and obtain the first lower bounds on their communication costs. In the sequential case, where the processor has a fast memory of size M , too small to store three n -by- n matrices, the lower bound on the number of words moved between fast and slow memory is, for a large class of matrix multiplication algorithms, Ω( ( n /√ M ) ω 0 · M ), where ω 0 is the exponent in the arithmetic count (e.g., ω 0 = lg 7 for Strassen, and ω 0 = 3 for conventional matrix multiplication). With p parallel processors, each with fast memory of size M , the lower bound is asymptotically lower by a factor of p . These bounds are attainable both for sequential and for parallel algorithms and hence optimal.

A Modified Split-Radix FFT With Fewer Arithmetic Operations

Recent results by Van Buskirk have broken the record set by Yavne in 1968 for the lowest exact count of real additions and multiplications to compute a power-of-two discrete Fourier transform (DFT). Here, we present a simple recursive modification of the split-radix algorithm that computes the DFT with asymptotically about 6% fewer operations than Yavne, matching the count achieved by Van Buskirk's program-generation framework. We also discuss the application of our algorithm to real-data and real-symmetric (discrete cosine) transforms, where we are again able to achieve lower arithmetic counts than previously published algorithms

A Fast Algorithm for Periodic Structures in Vibration

Periodic structures are popular in engineering applications for economical and architectural reasons. A fast method for calculating the natural vibration of periodic structures having identical constituents (substructures) with general boundary conditions is introduced. The number of arithmetic counts is proportional to PlogP, instead of P2 in the conventional method with band solver, where P is the number of substructures involved. Because of the fact that the general boundary conditions have been considered, a periodic structure can be represented by its condensed matrices associated with the boundary stations to form a superstructure which can be connected to other structures. Four illustrative examples are given.