Infinite Variant Research Articles

Infinite quantum permutations

We define and study quantum permutations of infinite sets. This leads to discrete quantum groups which can be viewed as infinite variants of the quantum permutation groups introduced by Wang. More precisely, the resulting quantum groups encode universal quantum symmetries of the underlying sets among all discrete quantum groups.We also discuss quantum automorphisms of infinite graphs, including some examples and open problems regarding both the existence and non-existence of quantum symmetries in this setting.

82. Enabling large scale precision oncology research with a new standard for genomic variants: OMOP Genomic

Precision Oncology requires the application of genomic variants to epidemiological methods used in clinical research. Variants are detected through traditional immunohistochemistry, microarray or PCR panels, or large-scale next generation sequencing, either for purposes of clinical care or understanding of biological processes of carcinogenesis and cancer treatment. However, we see few variants used in standard epidemiological research in the context of rich phenotypes: longitudinal observational patient data of treatment, progression and survival. The reason for this lies in the incompatibility of data representations and methods. In a typical observational study, cohorts, exposures, and outcomes are defined through clinical events encoded by predefined concepts. Epidemiological methods use this closed system model to measure the statistical change in rates, which are calculated with the denominator of all possible clinical states. The same is very challenging for genomic variants, which work like open systems: The representation of possibly infinite variants, whether previously observed or not. The Oncology Working Group of OHDSI has been working on the development of a canonical, comprehensive, and non-redundant representation of genomic variants that are clinically relevant for cancer. The resulting reference list, provided as part of the OMOP Standardized Vocabularies, is called 'OMOP Genomic'. It was constructed by consolidating genomic variants from public somatic cancer variant knowledgebases and contains +95,000 variations from 575 cancer genes. Assessment of the current gap, integration of other genomic knowledgebases are essential to improve the coverage of important and clinically relevant mutations implicated in cancer. Precision Oncology requires the application of genomic variants to epidemiological methods used in clinical research. Variants are detected through traditional immunohistochemistry, microarray or PCR panels, or large-scale next generation sequencing, either for purposes of clinical care or understanding of biological processes of carcinogenesis and cancer treatment. However, we see few variants used in standard epidemiological research in the context of rich phenotypes: longitudinal observational patient data of treatment, progression and survival. The reason for this lies in the incompatibility of data representations and methods. In a typical observational study, cohorts, exposures, and outcomes are defined through clinical events encoded by predefined concepts. Epidemiological methods use this closed system model to measure the statistical change in rates, which are calculated with the denominator of all possible clinical states. The same is very challenging for genomic variants, which work like open systems: The representation of possibly infinite variants, whether previously observed or not. The Oncology Working Group of OHDSI has been working on the development of a canonical, comprehensive, and non-redundant representation of genomic variants that are clinically relevant for cancer. The resulting reference list, provided as part of the OMOP Standardized Vocabularies, is called 'OMOP Genomic'. It was constructed by consolidating genomic variants from public somatic cancer variant knowledgebases and contains +95,000 variations from 575 cancer genes. Assessment of the current gap, integration of other genomic knowledgebases are essential to improve the coverage of important and clinically relevant mutations implicated in cancer.

HYPERSURFACE SUPPORT FOR NONCOMMUTATIVE COMPLETE INTERSECTIONS

AbstractWe introduce an infinite variant of hypersurface support for finite-dimensional, noncommutative complete intersections. We show that hypersurface support defines a support theory for the big singularity category $\operatorname {Sing}(R)$ , and that the support of an object in $\operatorname {Sing}(R)$ vanishes if and only if the object itself vanishes. Our work is inspired by Avramov and Buchweitz’ support theory for (commutative) local complete intersections. In the companion piece [27], we employ hypersurface support for infinite-dimensional modules, and the results of the present paper, to classify thick ideals in stable categories for a number of families of finite-dimensional Hopf algebras.

On infinite variants of De Morgan law in locale theory

A locale, being a complete Heyting algebra, satisfies De Morgan law (a∨b)⁎=a⁎∧b⁎ for pseudocomplements. The dual De Morgan law (a∧b)⁎=a⁎∨b⁎ (here referred to as the second De Morgan law) is equivalent to, among other conditions, (a∨b)⁎⁎=a⁎⁎∨b⁎⁎, and characterizes the class of extremally disconnected locales. This paper presents a study of the subclasses of extremally disconnected locales determined by the infinite versions of the second De Morgan law and its equivalents.

The Ray Method and Identification Problems for Equations of the Theory of Elasticity

Using algebraic-analytic methods, we establish numerous connections, in finite and infinite variants, between amplitudes, coefficients, and source functions of dynamical systems of elasticity theory.

Approximation of the continuous functions on $$l_{p}$$ spaces with p an even natural number

Let p be an even natural number, $$K\subset l_{p}$$ a compact nonempty set and $$L_{n}:C\left( K\right) \rightarrow C\left( K\right) $$ a sequence of positive linear operators. We prove that, under suitable assumptions, for every $$f\in C\left( K\right) $$, $$\lim \nolimits _{n\rightarrow \infty }L_{n}\left( f\right) =f$$ uniformly on K. As applications we give infinite variants of the Bernstein and Kantorovich theorems.

The finite intersection principle and genericity

AbstractWe show that a Δ02 Turing degree computes solutions to all computable instances of the finite intersection principle if and only if it computes a 1-generic degree. We also investigate finite and infinite variants of the principle.

Spatio-temporal learning with the online finite and infinite echo-state Gaussian processes.

Successful biological systems adapt to change. In this paper, we are principally concerned with adaptive systems that operate in environments where data arrives sequentially and is multivariate in nature, for example, sensory streams in robotic systems. We contribute two reservoir inspired methods: 1) the online echostate Gaussian process (OESGP) and 2) its infinite variant, the online infinite echostate Gaussian process (OIESGP) Both algorithms are iterative fixed-budget methods that learn from noisy time series. In particular, the OESGP combines the echo-state network with Bayesian online learning for Gaussian processes. Extending this to infinite reservoirs yields the OIESGP, which uses a novel recursive kernel with automatic relevance determination that enables spatial and temporal feature weighting. When fused with stochastic natural gradient descent, the kernel hyperparameters are iteratively adapted to better model the target system. Furthermore, insights into the underlying system can be gleamed from inspection of the resulting hyperparameters. Experiments on noisy benchmark problems (one-step prediction and system identification) demonstrate that our methods yield high accuracies relative to state-of-the-art methods, and standard kernels with sliding windows, particularly on problems with irrelevant dimensions. In addition, we describe two case studies in robotic learning-by-demonstration involving the Nao humanoid robot and the Assistive Robot Transport for Youngsters (ARTY) smart wheelchair.

Hereditary Uniserial Categories with Serre Duality

An abelian Krull-Schmidt category is said to be uniserial if the isomorphism classes of subobjects of a given indecomposable object form a linearly ordered poset. In this paper, we classify the hereditary uniserial categories with Serre duality. They fall into two types: the first type is given by the representations of the quiver A n with linear orientation (and infinite variants thereof), the second type by tubes (and an infinite variant). These last categories give a new class of hereditary categories with Serre duality, called big tubes.

The three-color and two-color Tantrix™ rotation puzzle problems are NP-complete via parsimonious reductions

Holzer and Holzer [M. Holzer, W. Holzer, Tantrix™ rotation puzzles are intractable, Discrete Applied Mathematics 144(3) (2004) 345–358] proved that the Tantrix™ rotation puzzle problem with four colors is NP-complete, and they showed that the infinite variant of this problem is undecidable. In this paper, we study the three-color and two-color Tantrix™ rotation puzzle problems (3-TRP and 2-TRP) and their variants. Restricting the number of allowed colors to three (respectively, to two) reduces the set of available Tantrix™ tiles from 56 to 14 (respectively, to 8). We prove that 3-TRP and 2-TRP are NP-complete, which answers a question raised by Holzer and Holzer [M. Holzer, W. Holzer, Tantrix™ rotation puzzles are intractable, Discrete Applied Mathematics 144(3) (2004) 345–358] in the affirmative. Since our reductions are parsimonious, it follows that the problems Unique-3-TRP and Unique-2-TRP are DP-complete under randomized reductions. We also show that the another-solution problems associated with 4-TRP, 3-TRP, and 2-TRP are NP-complete. Finally, we prove that the infinite variants of 3-TRP and 2-TRP are undecidable.

Computational Power of Infinite Quantum Parallelism

Recent works have independently suggested that quantum mechanics might permit procedures that fundamentally transcend the power of Turing Machines as well as of ‘standard’ Quantum Computers. These approaches rely on and indicate that quantum mechanics seems to support some infinite variant of classical parallel computing. We compare this new one with other attempts towards hypercomputation by separating (1) its %principal computing capabilities from (2) realizability issues. The first are shown to coincide with recursive enumerability; the second are considered in analogy to ‘existence’ in mathematical logic.

Scaling limits of the fuzzy sphere at one loop

We study the one loop dynamics of QFT on the fuzzy sphere and calculate the planar and nonplanar contributions to the two point function at one loop. We show that there is no UV/IR mixing on the fuzzy sphere. The fuzzy sphere is characterized by two moduli: a dimensionless parameter N and a dimensionful radius R. Different geometrical phases can obtained at different corners of the moduli space. In the limit of the commutative sphere, we find that the two point function is regular without UV/IR mixing; however quantization does not commute with the commutative limit, and a finite ``noncommutative anomaly'' survives in the commutative limit. In a different limit, the noncommutative plane R^2_theta is obtained, and the UV/IR mixing reappears. This provides an explanation of the UV/IR mixing as an infinite variant of the ``noncommutative anomaly''.

Taking It to the Limit: On Infinite Variants of NP-Complete Problems

We define infinite, recursive versions of NP optimization problems. For example,Max Cliquebecomes the question of whether a recursive graph contains an infinite clique. The present paper was motivated by trying to understand what makes some NP problems highly undecidable in the infinite case, while others remain on low levels of the arithmetical hierarchy. We prove two results; one enables using knowledge about the infinite case to yield implications to the finite case, and the other enables implications in the other direction. Moreover, taken together, the two results provide a method for proving (finitary) problems to be outside the syntactic classMaxNP, and, hence, outsideMaxSNP too, by showing that their infinite versions areΣ11-complete. We illustrate the technique with many examples, resulting ?in a large number of newΣ11-complete problems.