Density of type-dependent sets in Krull monoids with analytic structure

We describe structural and quantitative properties of type-dependent sets in monoids with suitable analytic structure, including simple analytic monoids, introduced by Kaczorowski (Semigroup Forum 94:532–555, 2017. 10.1007/s00233-016-9778-9), and formations, as defined by Geroldinger and Halter-Koch (Non-unique factorizations, Chapman and Hall, Boca Raton, 2006. 10.1201/9781420003208). We propose the notions of rank and degree to measure the size of a type-dependent set in structural terms. We also consider various notions of regularity of type-dependent sets, related to the analytic properties of their zeta functions, and obtain results on the counting functions of these sets.

Ethnomathematics Exploration: Offering Dance Performance (Makan Sirih) Ethnic Malay Deli North Sumatra

The purpose of this research is to explore a relationship between applied mathematics and a culture called ethnomathematics. The culture that will be used as the research topic is the performance culture of the Offering Dance (Makan Sirih) from the Deli Malay ethnic group in North Sumatra. This study is a research that uses descriptive qualitative research methods using an ethnographic approach that analyzes and describes a local culture based on facts obtained in the field. From the results of this research, it is revealed that, in the performance of the Offering Dance (Makan Sirih) in the Malay ethnic Deli of North Sumatra, there are various applications of mathematical concepts such as the concept of sets, counting, functions, and flat shapes. This can lead to new breakthroughs that underlie the formation of new mathematics learning designs for educational institutions. Along with this, this research also aims to improve the public's view of mathematics, that mathematics is a science that also has connections with all forms of activities in daily life.

Counting Homomorphisms to Trees Modulo a Prime

Many important graph-theoretic notions can be encoded as counting graph homomorphism problems, such as partition functions in statistical physics, in particular independent sets and colourings. In this article, we study the complexity of # p H OMS T O H , the problem of counting graph homomorphisms from an input graph to a graph H modulo a prime number p . Dyer and Greenhill proved a dichotomy stating that the tractability of non-modular counting graph homomorphisms depends on the structure of the target graph. Many intractable cases in non-modular counting become tractable in modular counting due to the common phenomenon of cancellation. In subsequent studies on counting modulo 2, however, the influence of the structure of H on the tractability was shown to persist, which yields similar dichotomies. Our main result states that for every tree H and every prime p the problem # p H OMS T O H is either polynomial time computable or # p P-complete. This relates to the conjecture of Faben and Jerrum stating that this dichotomy holds for every graph H when counting modulo 2. In contrast to previous results on modular counting, the tractable cases of # p H OMS T O H are essentially the same for all values of the modulo when H is a tree. To prove this result, we study the structural properties of a homomorphism. As an important interim result, our study yields a dichotomy for the problem of counting weighted independent sets in a bipartite graph modulo some prime p . These results are the first suggesting that such dichotomies hold not only for the modulo 2 case but also for the modular counting functions of all primes p .

Investigating Transformational Complexity: Counting Functions a Region Induces on Another in Elementary Cellular Automata

Over the years, the field of artificial life has attempted to capture significant properties of life in artificial systems. By measuring quantities within such complex systems, the hope is to capture the reasons for the explosion of complexity in living systems. A major effort has been in discrete dynamical systems such as cellular automata, where very few rules lead to high levels of complexity. In this paper, for every elementary cellular automaton, we count the number of ways a finite region can transform an enclosed finite region. We discuss the relation of this count to existing notions of controllability, physical universality, and constructor theory. Numerically, we find that particular sizes of surrounding regions have preferred sizes of enclosed regions on which they can induce more transformations. We also find three particularly powerful rules (90, 105, 150) from this perspective.

Fourier Interpolation and Time-Frequency Localization

We prove that under very mild conditions for any interpolation formula $$f(x) = \sum _{\lambda \in \Lambda } f(\lambda )a_\lambda (x) + \sum _{\mu \in M} {\hat{f}}(\mu )b_{\mu }(x)$$ f ( x ) = ∑ λ ∈ Λ f ( λ ) a λ ( x ) + ∑ μ ∈ M f ^ ( μ ) b μ ( x ) we have a lower bound for the counting functions $$n_\Lambda (R_1) + n_{M}(R_2) \ge 4R_1R_2 - C\log ^{2}(4R_1R_2)$$ n Λ ( R 1 ) + n M ( R 2 ) ≥ 4 R 1 R 2 - C log 2 ( 4 R 1 R 2 ) which very closely matches recent interpolation formulas of Radchenko and Viazovska and of Bondarenko, Radchenko and Seip.

Orbit Growth of Shift Spaces Induced by Bouquet Graphs and Dyck Shifts

For a discrete dynamical system, the prime orbit and Mertens’ orbit counting functions describe the growth of its closed orbits in a certain way. The asymptotic behaviours of these counting functions can be determined via Artin–Mazur zeta function of the system. Specifically, the existence of a non-vanishing meromorphic extension of the zeta function leads to certain asymptotic results. In this paper, we prove the asymptotic behaviours of the counting functions for a certain type of shift spaces induced by directed bouquet graphs and Dyck shifts. We call these shift spaces as the bouquet-Dyck shifts. Since their respective zeta function involves square roots of polynomials, the meromorphic extension is difficult to be obtained. To overcome this obstacle, we employ some theories on zeros of polynomials, including the well-known Eneström–Kakeya Theorem in complex analysis. Finally, the meromorphic extension will imply the desired asymptotic results.

The Schottky–Klein prime function and counting functions for Fenchel double crosses

We relate the classical nineteenth century Schottky–Klein function in complex analysis to a counting problem for pairs of geodesics in hyperbolic geometry studied by Fenchel. We then solve the counting problem using ideas from ergodic theory and thermodynamic formalism.

Weyl invariant Jacobi forms along Higgsing trees

Using topological string techniques, we compute BPS counting functions of 5d gauge theories which descend from 6d superconformal field theories upon circle compactification. Such theories are naturally organized in terms of nodes of Higgsing trees. We demonstrate that the specialization of the partition function as we move from the crown to the root of a tree is determined by homomorphisms between rings of Weyl invariant Jacobi forms. Our computations are made feasible by the fact that symmetry enhancements of the gauge theory which are manifest on the massless spectrum are inherited by the entire tower of BPS particles. In some cases, these symmetry enhancements have a nice relation to the 1-form symmetry of the associated gauge theory.

Validity of a Smart-Glasses-Based Step-Count Measure during Simulated Free-Living Conditions

Step counting represents a valuable approach to monitor the amount of daily physical activity. The feet, wrist and trunk have been demonstrated as the ideal locations to automatically detect the number of steps through body-worn devices (i.e., step counters). Key features of such devices are high usability, practicality and unobtrusiveness. Therefore, the opportunity to integrate step-counting functions in daily worn accessories represents one of the recent and most important challenges. In this context, the present study aimed to investigate the validity of a smart-glasses-based step-counter measure by comparing their performances against the most popular commercial step counters. To this purpose, smart glasses data from 26 healthy subjects performing simulated free-living walking conditions along a predefined path were collected. Reference measures from inertial sensors mounted on the subjects’ ankles and data from commercial (waist- and wrists-worn) step counters were acquired during the tests. The results showed an overall percentage error of 1%. In conclusion, the proposed smart glasses could be considered an accurate step counter, showing performances comparable to the most common commercial step counters.

On the Quartic Residues and Their New Distribution Properties

In this paper, we use the analytic methods, the properties of the fourth-order characters, and the estimate for character sums to study the computational problems of one kind of special quartic residues modulo p, and give an exact calculation formula and asymptotic formula for their counting functions.