General Theory Research Articles

Strong error bounds for Trotter and strang-splittings and their implications for quantum chemistry

Efficient error estimates for the Trotter product formula are central in quantum computing, mathematical physics, and numerical simulations. However, the Trotter error's dependency on the input state and its application to unbounded operators remains unclear. Here, we present a general theory for error estimation, including higher-order product formulas, with explicit input state dependency. Our approach overcomes two limitations of the existing operator-norm estimates in the literature. First, previous bounds are too pessimistic as they quantify the worst-case scenario. Second, previous bounds become trivial for unbounded operators and cannot be applied to a wide class of Trotter scenarios, including atomic and molecular Hamiltonians. Our method enables analytical treatment of Trotter errors in chemistry simulations, illustrated through a case study on the hydrogen atom. Our findings reveal the following: (i) for states with fat-tailed energy distribution, such as low-angular-momentum states of the hydrogen atom, the Trotter error scales worse than expected (sublinearly) in the number of Trotter steps; (ii) certain states do not admit an advantage in the scaling from higher-order Trotterization and, thus, the higher-order Trotter hierarchy breaks down for these states, including the hydrogen atom's ground state; (iii) the scaling of higher-order Trotter bounds might depend on the order of the Hamiltonians in the Trotter product for states with fat-tailed energy distribution. Physically, the enlarged Trotter error is caused by the atom's ionization due to the Trotter dynamics. Mathematically, we find that certain domain conditions are not satisfied by some states so higher moments of the potential and kinetic energies diverge. Our analytical error analysis agrees with numerical simulations, indicating that we can estimate the state-dependent Trotter error scaling genuinely. Published by the American Physical Society 2024

General theory of cavity-mediated interactions between low-energy matter excitations.

The manipulation of low-energy matter properties such as superconductivity, ferromagnetism, and ferroelectricity via cavity quantum electrodynamics engineering has been suggested as a way to enhance these many-body collective phenomena. In this work, we investigate the effective interactions between low-energy matter excitations induced by the off-resonant coupling with cavity electromagnetic modes. We extend a previous work by going beyond the dipole approximation accounting for the full polarization and magnetization densities of matter. We further include the often neglected diamagnetic interaction and, for the cavity, we consider general linear absorbing media with possibly non-local and non-reciprocal response. We demonstrate that, even in this general scenario, the effective cavity-induced interactions between the matter degrees of freedom are of electrostatic and magnetostatic nature. This confirms the necessity of a multimode description for cavity engineering of matter systems where the low-energy assumption holds. Our findings provide a theoretical framework for studying the influence of general optical environments on extended low-energy matter excitations.

Soft-photon theorem for pion-proton elastic scattering revisited

We discuss the reactions πp→πp and πp→πpγ from a general quantum field theory (QFT) point of view, describing these reactions in QCD and lowest relevant order of electromagnetism. We consider the pion-proton elastic scattering both off shell and on shell. The on-shell amplitudes for π±p→π±p scattering are described by two invariant amplitudes, while the off-shell amplitudes contain eight invariant amplitudes. We study the photon emission amplitudes in the soft-photon limit where the center-of-mass photon energy ω→0. The Laurent expansion in ω of the π±p→π±pγ amplitudes is considered and the terms of the orders ω−1 and ω0 are derived. These terms can be expressed by the on-shell invariant amplitudes and their partial derivatives with respect to s and t. The pole term ∝ω−1 in the amplitudes corresponds to Weinberg’s soft-photon theorem and is well known from the literature. We derive the next-to-leading term ∝ω0 using only rigorous methods of QFT. We give the relation of the Laurent series for π0p→π0pγ and Low’s soft-photon theorem. Our formulas for the amplitudes in the limit ω→0 are valid for photon momentum k satisfying k2≥0, k0=ω≥0, that is, for both real and virtual photons. Here we consider a limit where with ω→0 we have also k2→0. We discuss the behavior of the corresponding cross sections for π−p→π−pγ with respect to ω for ω→0. We consider cross sections for unpolarized as well as polarized protons in the initial and final states. Published by the American Physical Society 2024

Multilevel modeling and control of dynamic systems.

The underlying changes of complex dynamic systems are affected by numerous highly interdependent components with nonlinear interactions and cannot be suitably predicted by merely analyzing their individual parts. Recent technological developments (reducing the size of actuators and sensors, increasing speed and productivity) make it possible to monitor and handle systems on time, even during transient processes. In many practical applications, to effectively study rapidly evolving systems, their structure has to be examined at three levels: Micro-Scale (Short-Term), Meso-Scale (Intermediate-Term), and Macro-Scale (Long-Term). This multiscale framework provides a powerful tool for understanding and managing the intricate dynamics of complex systems. Macro-scale and micro-scale approaches are conventionally utilized in general system modeling and control theory, especially within the fields of networked control and multi-agent systems. By leveraging the meso-scale perspective, it is possible to balance detailed component analysis with overarching system behaviors, thereby enhancing the efficiency and effectiveness of operational strategies in complex systems. This paper presents and discusses a formal meso-level depiction of dynamics and control, assuming a finite number of attractors form and remain stable over time progress. Integral characteristics and dynamics of clusters are defined, simplifying control synthesis while maintaining problem formulation. A significant challenge is the model error due to changing cluster structures linked to randomized estimation and optimization algorithms valid under unknown disturbances. To demonstrate the practicality of this framework, the approach is applied to control tasks of a group of robots, leading to scalable and effective control strategies.

A noetic account of explanation in mathematics

Abstract We defend a noetic account of intramathematical explanation. On this view, a piece of mathematics is explanatory just in case it produces understanding of an appropriate type. We motivate the view by presenting some appealing features of noeticism. We then discuss and criticize the most prominent extant version of noeticism, due to Inglis and Mejía Ramos, which identifies explanatory understanding with the possession of well-organized cognitive schemas. Finally, we present a novel noetic account. On our view, explanatory understanding arises from meeting specific explanatory objectives. We defend a cluster-concept account of explanatory objectives and identify four important subfamilies within the relevant network of resemblance relations. The resulting view is objectivist (in the sense that it takes explanatory success to be a matter of observer-independent fact), broader in scope than why-question-based accounts, compatible with empirical findings on experts’ explanatory judgments, and capable of generalizing (with appropriate provisos) to scientific explanation as a whole. It thus fulfills Friedman’s half-century-old demand for a general and objectivist theory, which accounts for the link between explanation and understanding.

General theory for infernal points in non-Hermitian systems

General theory for infernal points in non-Hermitian systems

THE KEY DIRECTIONS FOR IMPROVING THE FINANCIAL MANAGEMENT OF THE TAX SYSTEM IN THE REPUBLIC OF ARMENIA

The transition to market relations involves addressing numerous challenges, and resolving these challenges is essential for creating the necessary preconditions for the further development of the market economy in the Republic of Armenia (RA). This includes the implementation of radical transformations in the country's financial, monetary, banking, fiscal, social, structural, and other systems. During the reforms of the financial system in the Republic of Armenia, special attention has been given to the creation and ongoing improvement of an effective tax system. In this regard, the RA State Revenue Committee, as the tax authority, has served as a state government body responsible for tax administration for more than 30 years. Over this period, it has accumulated significant experience and, in line with contemporary developments, has implemented various infrastructures and information systems. According to the author, the financial management of the RA tax system falls within the general theory of management, and the functions of financial management, such as financial planning, analysis, forecasting, and control, are adequately reflected in this context. The article emphasizes the conceptual foundations of program budgeting and accounting, which ensure a practical, close connection between the financial management of the RA tax system and accounting, budgeting, and the financial flows of the tax system and their related transactions. This connection leads to the effective management of the assets and liabilities of the RA tax system.

Modeling Hydraulic Fracture Entering Stress Barrier: Theory and Practical Recommendations

Numerical modeling of hydraulic fracturing is complicated when a fracture reaches a stress barrier. For high barriers, it may require changing the computational scheme. In view of the strong influence of stress barriers on the final footprint and opening of a hydraulic fracture, for decades, their modeling has been the subject of special investigations. Actually, classical models of propagation within a pay layer with impenetrable boundaries referred to the case of extremely high-stress contrast in neighbor layers. Further improvements tended to account for the fracture growth in these layers by including stress contrasts as input parameters of a model. This tendency resulted in the suggestion and successive enhancements of pseudo-three-dimensional models. All of them have used stress intensity factors (SIFs) to characterize the combined resistance caused by stress contrasts, material strength, and fluid viscosity. Specifically, the SIFs served to formulate the conditions that control the front penetration into a neighbor layer. This key concept presents the background of our research. Despite examples of modeling propagation through barriers, there is no general theory clarifying when and why conventional schemes may become inefficient and how to overcome computational difficulties. This paper presents the theory and practical recommendations following it. We start with the definition of the barrier intensity, which exposes that the barrier strength may change from zero for contrast-free propagation to infinity for channelized propagation. The analysis reveals two types of computational difficulties caused by spatial discretization as follows: (i) general, arising for fine grids and aggravated by a barrier, and (ii) specific, caused entirely by a strong barrier. The asymptotic approach, which avoids spatial discretization, is suggested. It is illustrated by solving benchmark problems for barriers of arbitrary intensity. The analysis distinguishes three typical stages of the fracture penetration into a barrier and provides theoretical values of the Nolte–Smith slope parameter and arrest time as functions of the barrier intensity. Special analysis establishes the accuracy and bounds of the asymptotic approach. It appears that the approach provides physically significant and accurate results for fracture penetration into high, intermediate, and even weak stress barriers. On this basis, simple practical recommendations are given for modeling hydraulic fractures in rocks with stress barriers. The recommendations may be promptly implemented in any program using spatial discretization to model fracture propagation.

Schlick on intuition and prediction

ABSTRACT Textor's The Disappearance of the Soul and the Turn Against Metaphysics examines the voluntarist background of Schlick's epistemology, including his conception of knowledge as essentially involving judgements that relate at least two terms, and his connected objection against according intuition epistemic status. Textor interprets Schlick's conception of intuition in light of Schopenhauer's distinction between ordinary and extraordinary cognition. Thus Textor argues that Schlick takes intuition to be a form of “steady contemplation” (Disappearance, 348) of an object that is “either a universal or a particular” (332). I suggest an alternative interpretation, according to which Schlick takes intuition to be an immersion into the contents of the stream of consciousness, which he claims to be a formless “Heraclitian flux” (General Theory, 31, 156). This interpretation allows for a defence, against Textor's objections, of Schlick's thesis that the aim of knowledge is prediction. If the kind of flux given in pure intuition is genuinely ‘Heraclitian’ – that is to say, if the stream of consciousness is ever-changing and unrepeatable – then it is simply impossible for “the devotee of intuition” (General Theory, 150) to predict its course. By contrast, the process of re-cognition that constitutes knowledge is intimately tied to prediction.

Impurities with a cusp: general theory and 3d Ising

In CFTs, the partition function of a line defect with a cusp depends logarithmically on the size of the line with an angle-dependent coefficient: the cusp anomalous dimension. In the first part of this work, we study the general properties of the cusp anomalous dimension. We relate the small cusp angle limit to the effective field theory of defect fusion, making predictions for the first couple of terms in the expansion. Using a concavity property of the cusp anomalous dimension we argue that the Casimir energy between a line defect and its orientation reversal is always negative (“opposites attract”). We use these results to determine the fusion algebra of Wilson lines in N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 SYM as well as pinning field defects in the Wilson-Fisher fixed points. In the second part of the paper we obtain nonperturbative numerical results for the cusp anomalous dimension of pinning field defects in the Ising model in d = 3, using the recently developed fuzzy-sphere regularization. We also compute the pinning field cusp anomalous dimension in the O(N) model at one-loop in the ε-expansion. Our results are in agreement with the general theory developed in the first part of the work, and we make several predictions for impurities in magnets.

Color-kinematics duality for nonlinear sigma models with nonsymmetric cosets

We propose an extension of the color-kinematics duality for nonlinear sigma models with nonymmetric cosets. The role of color is replaced with curvatures in field space. The duality between curvature and kinematics allows for a new double copy: the double copy of the nonlinear sigma model for nonsymmetric cosets is the general Galileon theory. We provide evidence for this new double copy of geometry. Published by the American Physical Society 2024

Global bifurcation for monotone fronts of elliptic equations

We present two results on global continuation of monotone front-type solutions to elliptic PDEs posed on infinite cylinders. This is done under quite general assumptions, and in particular applies even to fully nonlinear equations as well as quasilinear problems with transmission boundary conditions. Our approach is rooted in the analytic global bifurcation theory of Dancer (1973) and Buffoni–Toland (2003), but extending it to unbounded domains requires contending with new potential limiting behavior relating to loss of compactness. We obtain an exhaustive set of alternatives for the global behavior of the solution curve that is sharp, with each possibility having a direct analogue in the bifurcation theory of second-order ODEs.As a major application of the general theory, we construct global families of internal hydrodynamic bores. These are traveling front solutions of the full two-phase Euler equation in two dimensions. The fluids are confined to a channel that is bounded above and below by rigid walls, with incompressible and irrotational flow in each layer. Small-amplitude fronts for this system have been obtained by several authors. We give the first large-amplitude result in the form of continuous curves of elevation and depression bores. Following the elevation curve to its extreme, we find waves whose interfaces either overturn (develop a vertical tangent) or become exceptionally singular in that the flow in both layers degenerates at a single point on the boundary. For the curve of depression waves, we prove that either the interface overturns or it comes into contact with the upper wall.

Momentum shift and on-shell recursion relation for electroweak theory

We study the all-line transverse (ALT) shift which we developed for on-shell recursion of amplitudes for particles of any mass. We discuss the validity of the shift for general theories of spin ≤1, and illustrate the connection between Ward identity and constructibility for massive spin-1 amplitude under the ALT shift. We apply the shift to the electroweak theory, and various four-point scattering amplitudes among electroweak gauge bosons and fermions are constructed. We show explicitly that the four-point gauge boson contact terms in massive electroweak theory automatically arise after recursive construction, independent of UV completion, and they automatically cancel the terms growing as (energy)4 at high energy. We explore UV completion of the electroweak theory that cancels the remaining (energy)2 terms and impose unitarity requirements to constrain additional couplings. The ALT shift framework allows consistent treatment in dealing with contact term ambiguities for renormalizable massive and massless theories, which we show can be useful in studying real-world amplitudes with massive spinors. Published by the American Physical Society 2024

On some Airy sheaves of Laurent type

We study certain one-parameter families of exponential sums of Airy–Laurent type. Their general theory was developed in Katz and Tiep (Airy sheaves of Laurent type: an introduction, https://web.math.princeton.edu/~nmk/kt31_11sept.pdf). In the present paper, we make use of that general theory to compute monodromy groups in some particularly simple families (in the sense of “simple to remember"), realizing Weyl groups of type E6\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$E_6$$\\end{document} and E8\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$E_8$$\\end{document}.

Pythagoreanism and Modern Models of Cosmology

Turning to the accumulated philosophical knowledge, the scientist almost always discovers that there are ideas in it which are far ahead of their time, but which can help in solving scientific problems of quite another historical epoch. In this respect one of the most revealing examples are associated with the discovery of heliocentrism and the General Theory of Relativity. Philosophical ideas and principles immanent in one or another particular science and serving for the solution of some problem of it form the very philosophical foundations of the given science. These, in turn, are by no means identical with the whole body of knowledge which philosophy creates. By reflecting on the foundations of science and turning to the analysis of its arbitrary forms – namely cosmology – philosophical reflection elaborates ideas and principles that can play an essential role in a certain stage of scientific inquiry.

Einstein’s spacetime curvature and gravitational waves: Elucidating the gradient flow of scalar invariants, variations in the Ricci scalar, and gravitational wave propagation

In the general theory of relativity initially rationalized by Einstein, the gravity is considered as an occurrence ensuing from the spacetime curvature, which is in turn triggered by the presence of mass. This article provides a detailed analysis of spacetime curvature and gravitational waves, enhancing our understanding of general relativity and astrophysics. Using both analytical and numerical methods, we examined the gradient flow of scalar invariants, variations in the Ricci scalar, and gravitational wave propagation. New findings include the identification of unique curvature patterns around rotating black holes, a detailed comparison of gradient flow behavior in different spacetime geometries, and the discovery of a correlation between scalar invariant fluctuations and gravitational wave amplitudes. These findings validate numerical techniques and reveal how different parameters affect gravitational wave characteristics. While simplified models are used, the present study offers a robust framework for future research. Our work aligns with previous studies but also reveals unique aspects of wave behavior, with implications for quantum gravity and cosmology.

Effective Kähler and auxiliary field potentials for chiral superfield models at three loops

I obtain the effective Kähler potential at three-loop order for a general renormalizable supersymmetric theory containing only chiral supermultiplets. The three-loop contribution is remarkably simple, consisting of only four terms involving three distinct renormalized master integrals. In the case of the Wess-Zumino model with a single chiral superfield, I also obtain the effective auxiliary field potential at three-loop order, extending previous results at one-loop order. The method used is inferential, relying on existing knowledge of the ordinary scalar effective potential. Published by the American Physical Society 2024

A Framework for Understanding Credibility: What Makes Credible Messengers “Credible” in a New York City–Based Sample of Gun Violence Intervention Programs?

The use of credible messengers to deliver violence reduction programs has spread rapidly since it was first used in a criminal justice context by Cure Violence in 1999. However, there is no clear conceptualization of what a credible messenger is, aside from vague definitions contained in the evaluation literature. We provide a “broad strokes” framework for developing a general theory of credibility that might extend from criminal justice across disciplines. The framework also provides the basis for developing a middle theory on credibility within the area of violence intervention. We demonstrate how this might be done, using the theoretical framework to explore a set of 22 interviews with credible messenger staff.

Exposure to Violence at Home and Bullying in School: A Test of an Additional Layer of General Strain Theory

ABSTRACT School bullying continues to be a major concern in Trinidad and Tobago. Despite its prevalence and governmental efforts to reduce bullying, relatively few studies have examined the relationship between exposure to violence at home and school bullying in this nation. This study intends to examine the relationship between exposure to violence at home, anger, school culture toward bullying, and school bullying within the context of General Strain Theory. Using data collected from 1,248 students in ten primary schools in Trinidad, this study found that exposure to violence at home had significant influences on bullying in school. Further, anger significantly mediated the relationship between exposure to violence at home and school bullying while the moderating effects of anger and school culture favorable toward bullying were statistically significant.

Which Leadership Traits Matter Most? An Activity in Leadership Development

Leadership is widely associated with an array of attributes. The variety of attributes present both advantages and disadvantages. On one hand, attributes are appealing and easy to recognize in leadership. On the other hand, the number of attributes and which ones matter the most may be overwhelming. Grounded in experiential learning and situational interest, we present an implicit leadership activity with two main goals. First, we provide examples of attributes associated with many effective leadership prototypes. And second, we ask students to reflect on these attributes themselves and with others. The exercise is ideal for undergraduate introduction to leadership in general or implicit leadership theory. We provide instructions and resources to tailor the exercise to the needs of the instructor.