Fundamental Objects Research Articles

Logarithmic Vector Fields and Freeness of Divisors and Arrangements: New perspectives and applications

The central topic of the workshop was the notion of logarithmic vector fields along a divisor in a smooth complex analytic or algebraic variety, i.e., the vector fields on the ambient variety tangent to the divisor. Following their introduction by K. Saito for the purpose of studying the universal unfolding of an isolated singularity, this fundamental object has been the focus of studies in a wide range of mathematical fields such as algebra, algebraic geometry, singularity theory, root systems, (geometric) representation theory, combinatorics, (toric) topology, or symplectic geometry. In the last few years the logarithmic vector field approach has seen some unexpected and striking advances and deep applications. The aim of the workshop was to provide reports and to share these various new developments in the field.

Stabilizer rank and higher-order Fourier analysis

We establish a link between stabilizer states, stabilizer rank, and higher-order Fourier analysis – a still-developing area of mathematics that grew out of Gowers's celebrated Fourier-analytic proof of Szemerédi's theorem \cite{gowers1998new}. We observe that n-qudit stabilizer states are so-called nonclassical quadratic phase functions (defined on affine subspaces of Fpn where p is the dimension of the qudit) which are fundamental objects in higher-order Fourier analysis. This allows us to import tools from this theory to analyze the stabilizer rank of quantum states. Quite recently, in \cite{peleg2021lower} it was shown that the n-qubit magic state has stabilizer rank Ω(n). Here we show that the qudit analog of the n-qubit magic state has stabilizer rank Ω(n), generalizing their result to qudits of any prime dimension. Our proof techniques use explicitly tools from higher-order Fourier analysis. We believe this example motivates the further exploration of applications of higher-order Fourier analysis in quantum information theory.

Exact Distribution-Free Hypothesis Tests for the Regression Function of Binary Classification via Conditional Kernel Mean Embeddings

In this paper we suggest two statistical hypothesis tests for the regression function of binary classification based on conditional kernel mean embeddings. The regression function is a fundamental object in classification as it determines both the Bayes optimal classifier and the misclassification probabilities. A resampling based framework is presented and combined with consistent point estimators of the conditional kernel mean map, in order to construct distribution-free hypothesis tests. These tests are introduced in a flexible manner allowing us to control the exact probability of type I error for any sample size. We also prove that both proposed techniques are consistent under weak statistical assumptions, i.e., the type II error probabilities pointwise converge to zero.

Поля фундаментальных и охваченных геометрических объектов регулярной гиперполосы с центральным оснащением проективного пространства

The study of hyperbands and their generalizations in spaces with dif­ferent fundamental groups is of great interest in connection with numer­ous applications in mathematics and physics. In this paper, we study a special class of hyperbands, i. e., centrally equipped hyperbands. A hy­perband Hm (m ≥ 2) is said to be centrally rigged if the rigging lines in the normals of the 1st kind of the base surface pass through one (the center of the rigging). The article gives a task of a centrally equipped hyperband in the 1st order frame. A sequence of fundamental geometric objects of a hyperstrip with central framing is constructed. An existence theorem for a hyperband with a central framing is proved. It is proved that a hyperstrip with central framing and framing in the sense of Cartan induces a projective connec­tion obtained by projection, where the projection center at each point is the Cartan plane. The spans of the components of the curvature-torsion tensor of the constructed connection are found.

Discursos litúrgicos en piedra y metal. De los altares de Oviedo, Sahagún y Compostela a los de Mondoñedo, Ribas de Sil y O Incio

In the renewal of visual culture in Romanesque times, liturgical furniture played a fundamental role, with the altar being the most privileged figurative space, as a fundamental cultic object for the configuration of the new ritual spaces. With this work we propose a review of some of the main Romanesque altars in the region (Santiago de Compostela, Oviedo, San Esteban de Ribas de Sil, etc.) in order to analyse the value of the images they reproduce in relation to the new vectors of Romanesque culture.

ONTOLOGY OF LOOPED QUANTUM GRAVITY. HINGES

Some ontological aspects of the theory of loop quantum gravity are considered. First of all, its possible nature of one of the fundamental objects of theory - the quantum loop - is discussed. In particular, the possible material, geometric and instrumentalist interpretations of this object are briefly considered.

CRISPR System-Linked Self-Assembling Nanoplatforms for Inspection and Screening of Gastric Cancer Stem Cells.

Cancer stem cells (CSCs) possess a high degree of plasticity, constituting a formidable challenge to identify and screen CSCs in situ with outstanding specificity and sensitivity. To overcome this limitation, a self-assembled heterodimer consisting of clustered regularly interspaced short palindromic repeats/Cas12a (named A-CCA) linkage is designed for in situ identification and screening of gastric CSCs (GCSCs) from gastric cancer cells (GCCs). In this system, the editable character of crRNA performs recognition of dual-targets in GCSCs, effectively boosting the specificity of identification, while the enzymatic reaction of Cas12a contributes meaningfully to the sensitivity of sensing, enabling in situ examination and screening of GCSCs. Specifically, the A-CCA nanoplatforms hybridized with ABCG 2 and ABCB 1 overexpress in GCSCs, which can generate heterodimers and simultaneously restore the function of trans-cleavage. At this time, the asymmetry of the heterodimer causes a circular dichroism signal, which together with the recovered fluorescence signal form a dual-signals output system that can further ensure the precision of screening GCSC. Therefore, fluorescence-enhanced GCSCs can be sorted out from GCCs by flow cytometry. Furthermore, GCSCs screened by this assay possess extremely aggressive tumorigenic efficiency, providing a fundamental research object for further developing CSC targeted drugs in vivo.

Geometric approximation of the sphere by triangular polynomial spline patches

A sphere is a fundamental geometric object widely used in (computer aided) geometric design. It possesses rational parameterizations but no parametric polynomial parameterization exists. The present study provides an approach to the optimal approximation of equilateral spherical triangles by parametric polynomial patches if the measure of quality is the (simplified) radial error. As a consequence, optimal approximations of the unit sphere by parametric polynomial spline patches underlying on particular regular spherical triangulations arising from a tetrahedron, an octahedron and an icosahedron inscribed in the unit sphere are provided. Some low total degree spline patches with corresponding geometric smoothness are analyzed in detail and several numerical examples are shown confirming the quality of approximants.

Evaluating the advantage of adaptive strategies for quantum channel distinguishability

Recently, the resource theory of asymmetric distinguishability for quantum strategies was introduced by [Wang et al., Phys. Rev. Research 1, 033169 (2019)]. The fundamental objects in the resource theory are pairs of quantum strategies, which are generalizations of quantum channels that provide a framework to describe an arbitrary quantum interaction. In the present paper, we provide semi-definite program characterizations of the one-shot operational quantities in this resource theory. We then apply these semi-definite programs to study the advantage conferred by adaptive strategies in discrimination and distinguishability distillation of generalized amplitude damping channels. We find that there are significant gaps between what can be accomplished with an adaptive strategy versus a non-adaptive strategy.

A gentle introduction to the non-equilibrium physics of trajectories: Theory, algorithms, and biomolecular applications

Despite the importance of non-equilibrium statistical mechanics in modern physics and related fields, the topic is often omitted from undergraduate and core-graduate curricula. Key aspects of non-equilibrium physics, however, can be understood with a minimum of formalism based on a rigorous trajectory picture. The fundamental object is the ensemble of trajectories, a set of independent time-evolving systems, which easily can be visualized or simulated (e.g., for protein folding) and which can be analyzed rigorously in analogy to an ensemble of static system configurations. The trajectory picture provides a straightforward basis for understanding first-passage times, "mechanisms" in complex systems, and fundamental constraints on the apparent reversibility of complex processes. Trajectories make concrete the physics underlying the diffusion and Fokker-Planck partial differential equations. Last but not least, trajectory ensembles underpin some of the most important algorithms that have provided significant advances in biomolecular studies of protein conformational and binding processes.

Geometry, Coordinatization and Cardinality of the Rational Numbers from Physical Perspective

In this article, we will first discuss the completeness of real numbers in the context of an alternate definition of the straight line as a geometric continuum. According to this definition, points are not regarded as the basic constituents of a line segment and a line segment is considered to be a fundamental geometric object. This definition is in particular suitable to coordinatize different points on the straight line preserving the order properties of real numbers. Geometrically fundamental nature of line segments are required in physical theories like the string theory. We will construct a new topology suitable for this alternate definition of the straight line as a geometric continuum. We will discuss the cardinality of rational numbers in the later half of the article. We will first discuss what we do in an actual process of counting and define functions well-defined on the set of all positive integers. We will follow an alternate approach that depends on the Hausdorff topology of real numbers to demonstrate that the set of positive rationals can have a greater cardinality than the set of positive integers. This approach is more consistent with an actual act of counting. We will illustrate this aspect further using well-behaved functionals of convergent functions defined on the finite dimensional Cartezian products of the set of positive integers and non-negative integers. These are similar to the partition functions in statistical physics. This article indicates that the axiom of choice can be a better technique to prove theorems that use second-countability. This is important for the metrization theorems and physics of spacetime.

Optimization problem of external dynamic influences for the elimination of vibration phenomena in composite elastic building structures

The paper considers the actual task of construction engineering: the elimination of harmful (undesirable) vibration phenomena in composite metal building structures and bridge cable-stayed systems – complete damping and vibration of the bridge cable-stayed system entails the calming of the entire bridge structure. Similar problems arise when designing devices that lead to dynamic rest of network antenna structures. The presentation of the material is based on the main provisions of the modern theory of optimal control of distributed differential systems with network carriers. An extensive literature is devoted to the sanctification of the research of the mentioned problem, however, an exhaustive solution of the problems of controlling elastic structures by external influences of various types (distributed, boundary, starting, and point control) and a comparative analysis of their properties did not find their place in these studies. In addition, the ways of algorithmization of previously obtained results are not always specified. The paper uses a mathematical model describing small oscillations of a spatial string system. It is convenient to use such a system and approaches to the analysis of its mathematical model as a fairly simple fundamental object when considering a more complex object, which is an elastic system of more complex elements. The results of the work are presented in a form that makes it possible to formalize the computational procedure, which is convenient for algorithmizing the process of determining the solution, and therefore for the modern application of computer technology. It remains to add that the work indicates ways to transfer the obtained results to the case of analyzing optimal control problems with carriers on multidimensional network-like domains.

PARTS OF SPACETIME

Abstract Consider the following pair of theses: (i) all fundamental physical objects are spatiotemporal and (ii) all non-fundamental physical objects are ultimately composed of fundamental objects. Work on the physics of quantum gravity suggests that spacetime is a non-fundamental, emergent phenomenon and thus that thesis (i) is false. The fundamentals are non-spatiotemporal in nature. This paper will argue against (ii) on the grounds that non-fundamental spatiotemporal objects cannot be composed of fundamental non-spatiotemporal objects. So, assuming that spacetime is emergent, new metaphysical resources are needed to explain the relationship between the fundamental objects posited by quantum gravity and spacetime.

EDGE DOMINATION NUMBER AND THE NUMBER OF MINIMUM EDGE DOMINATING SETS IN PSEUDOFRACTAL SCALE-FREE WEB AND SIERPIŃSKI GASKET

As a fundamental research object, the minimum edge dominating set (MEDS) problem is of both theoretical and practical interest. However, determining the size of a MEDS and the number of all MEDSs in a general graph is NP-hard, and it thus makes sense to find special graphs for which the MEDS problem can be exactly solved. In this paper, we study analytically the MEDS problem in the pseudofractal scale-free web and the Sierpiński gasket with the same number of vertices and edges. For both graphs, we obtain exact expressions for the edge domination number, as well as recursive solutions to the number of distinct MEDSs. In the pseudofractal scale-free web, the edge domination number is one-ninth of the number of edges, which is three-fifths of the edge domination number of the Sierpiński gasket. Moreover, the number of all MEDSs in the pseudofractal scale-free web is also less than that corresponding to the Sierpiński gasket. We argue that the difference of the size and number of MEDSs between the two studied graphs lies in the scale-free topology.

Finite Cartan Graphs Attached to Nichols Algebras of Diagonal Type

Nichols algebras are fundamental objects in the construction of quantized enveloping algebras and in the classification of pointed Hopf algebras by the lifting method of Andruskiewitsch and Schneider. The structure of Cartan graphs can be attached to any Nichols algebras of diagonal type and plays an important role in the classification of Nichols algebras of diagonal type with a finite root system. In this paper, the main properties of all simply connected Cartan graphs attached to rank 6 Nichols algebras of diagonal type are determined. As an application, we obtain a subclass of rank 6 finite dimensional Nichols algebras of diagonal type.

Geometry and entanglement in the scattering matrix

A formulation of nucleon–nucleon scattering is developed in which the S-matrix, rather than an effective-field theory (EFT) action, is the fundamental object. Spacetime plays no role in this description: the S-matrix is a trajectory that moves between RG fixed points in a compact theory space defined by unitarity. This theory space has a natural operator definition, and a geometric embedding of the unitarity constraints in four-dimensional Euclidean space yields a flat torus, which serves as the stage on which the S-matrix propagates. Trajectories with vanishing entanglement are special geodesics between RG fixed points on the flat torus, while entanglement is driven by an external potential. The system of equations describing S-matrix trajectories is in general complicated, however the very-low-energy S-matrix –that appears at leading-order in the EFT description– possesses a UV/IR conformal invariance which renders the system of equations integrable, and completely determines the potential. In this geometric viewpoint, inelasticity is in correspondence with the radius of a three-dimensional hyperbolic space whose two-dimensional boundary is the flat torus. This space has a singularity at vanishing radius, corresponding to maximal violation of unitarity. The trajectory on the flat torus boundary can be explicitly constructed from a bulk trajectory with a quantifiable error, providing a simple example of a holographic quantum error correcting code.

Effects of quenching protocols based on parametric oscillators

We consider the problem of understanding the basic features displayed by quantum systems described by parametric oscillators whose time-dependent frequency parameter ω(t) varies continuously during evolution so to realize quenching protocols of different types. To this scope we focus on the case where ω(t)2 behaves like a Morse potential, up to possible sign reversion and translations in the (t,ω2) plane. We derive closed form solution for the time-dependent amplitude of quasi-normal modes, which is the very fundamental dynamical object entering the description of both classical and quantum parametric oscillators, and highlight its significant characteristics for distinctive cases arising based on the driving specifics. After doing so, we provide an insight on the way quantum states evolve by paying attention on the position–momentum Heisenberg uncertainty principle and the statistical aspects implied by second-order correlation functions over number-type states.

Tinygarden — A java package for testing properties of spanning trees

<h2>Abstract</h2> Spanning trees are fundamental objects in graph theory. The spanning tree set size of an arbitrary graph can be very large. This limitation discourages its analysis. However interesting patterns can emerge in small cases. In this article we introduce <i>tinygarden</i>, a java package for validating hypothesis, testing properties and discovering patterns from the spanning tree set of an arbitrary graph.

Symmetric distinguishability as a quantum resource

We develop a resource theory of symmetric distinguishability, the fundamental objects of which are elementary quantum information sources, i.e. sources that emit one of two possible quantum states with given prior probabilities. Such a source can be represented by a classical-quantum state of a composite system XA, corresponding to an ensemble of two quantum states, with X being classical and A being quantum. We study the resource theory for two different classes of free operations: (i) CPTP A , which consists of quantum channels acting only on A, and (ii) conditional doubly stochastic maps acting on XA. We introduce the notion of symmetric distinguishability of an elementary source and prove that it is a monotone under both these classes of free operations. We study the tasks of distillation and dilution of symmetric distinguishability, both in the one-shot and asymptotic regimes. We prove that in the asymptotic regime, the optimal rate of converting one elementary source to another is equal to the ratio of their quantum Chernoff divergences, under both these classes of free operations. This imparts a new operational interpretation to the quantum Chernoff divergence. We also obtain interesting operational interpretations of the Thompson metric, in the context of the dilution of symmetric distinguishability.

THE HOUSEHOLD FINANCIAL CONDUCT AND ITS IMPACT ON STABILITY FINANCIAL SYSTEM

The importance of maintaining Financial System Stability is the basis for economic sustainability. One of the pillars of national economic resilience is the role of the household sector as a fundamental object so that supply and demand reach an equilibrium point. Household financial behavior is closely related to income levels and household credit behavior towards Financial System Stability. Therefore, the aim of this study is to determine whether there is an effect of financial behavior in the household sector on financial system stability. Data collection using purposive sampling method was carried out using a questionnaire through the help of Google Forms application to 400 households in the Tangerang area. The analytical tool used is Structural Equation Modeling (SEM) with SmartPLS 3. 0 to explain the correlation between endogenous and exogenous variables. The loading factor results indicate that the value of financial behavior is 0.285, household income is 0.232 and household credit behavior is 0.229 has a significant effect on the financial stability system. Meanwhile, the value of financial behavior is 0.599 on household income and the value of financial behavior is 0.588 on household credit behavior which has a direct effect.