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The formation of a positive image of the educational institution provides an increase in the efficiency of its activities and provides an opportunity to meet the needs of stakeholders. The main purpose of the image formation is also to increase competitive advantage, attract investment, establish and expand partnerships. The development of teaching staff is an integral part of forming a positive image of the educational institution, as employees are the main carriers of the brand. The career of educators is a complex multifaceted process, due to the unity of internal (subjective) and external (objective) factors. Internal factors include the living conditions of the future specialist, and external - the peculiarities of career growth (attitude to the profession and awareness of its importance). The desire for self-improvement and self-education are important drivers for the formation of successful careers of employees of educational institutions, ensuring the expansion of their creative potential, cognitive interests, and the formation of a creative personality. Based on the conducted research on building an effective strategy for the development of the employee of the educational institution, each teacher was asked to conduct their SWOT analysis, which allows identifying his/her strengths and weaknesses, as well as external opportunities and threats. The results of the SWOT analysis enable the employee of the educational institution to discover his / her existing potential and to be ready for possible changes in the external environment, which is changeable and fleeting. The strategy of career growth of the teacher is developed, which includes the following stages: conducting SWOT analysis; improvement of professionally important features and qualities; postgraduate education; passing advanced training courses; participation in various pedagogical forums, seminars, conferences, training; research of the advanced pedagogical experience, in particular concerning the use of innovative technologies; constant improvement of scientific and methodical work; forming your portfolio and, as a result, achieving a new level of career growth. Career development leads to a fundamentally new way of life of a teacher - creative self-realization in the profession, which allows identifying their individual and professional capabilities.

In their 2011 paper on the AGT conjecture, Alba, Fateev, Litvinov and Tarnopolsky (AFLT) obtained a closed-form evaluation for a Selberg integral over the product of two Jack polynomials, thereby unifying the well-known Kadell and Hua--Kadell integrals. In this paper we use a variety of symmetric functions and symmetric function techniques to prove generalisations of the AFLT integral. These include (i) an $\mathrm{A}_n$ analogue of the AFLT integral, containing two Jack polynomials in the integrand; (ii) a generalisation of (i) for $\gamma=1$ (the Schur or GUE case), containing a product of $n+1$ Schur functions; (iii) an elliptic generalisation of the AFLT integral in which the role of the Jack polynomials is played by a pair of elliptic interpolation functions; (iv) an AFLT integral for Macdonald polynomials., Comment: 53 pages; v2 contains minor corrections and changes of notation

We employ the barycentric coordinate system to evaluate the inertia tensor of an arbitrary triangular plate of uniform mass distribution. We find that the physical quantities involving the computation are expressed in terms of a single master integral over barycentric coordinates. To expedite the computation in the barycentric coordinates, we employ Lagrange undetermined multipliers. The moment of inertia is expressed in terms of mass, barycentric coordinates of the pivot, and side lengths. The expression is unique and the most compact in comparison with popular expressions that are commonly used in the field of mechanical engineering. A master integral that is necessary to compute the integral over the triangle in the barycentric coordinate system and derivations of the barycentric coordinates of common triangle centers are provided in appendices. We expect that the barycentric coordinates are particularly efficient in computing physical quantities like the electrostatic potential of a triangular charge distribution. We also illustrate a practical experimental design that can be immediately applied to general-physics experiments.

Given a pair of regular quadratic forms over $\mathbb{Q}$ which are in the same genus and a finite set of primes $P$, we show that there is an effective way to determine a rational equivalence between these two quadratic forms which are integral over every prime in $P$. This answers one of the principal questions posed by Conway and Sloane in their book {\em Sphere packings, lattices and groups}, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Vol 290, Springer-Verlag, New York, 1999; page 402., 12 pages

We present a method to approximate, with controlled and arbitrarily small error, multiple intregrals over the unit cube $$[0,1]^{d}$$ by a single variable integral over [0, 1]. For this, we use the so called $$\delta $$ -uniform curves, which are a particular case of $$\alpha $$ -dense curves. Our main result improves and extends other existing methods on this subject.

In this paper the turnpike phenomenon is studied for problems of optimal control where both pointwise-in-time state and control constraints can appear. We assume that in the objective function, a tracking term appears that is given as an integral over the time-interval $$[0,\, T]$$ [ 0 , T ] and measures the distance to a desired stationary state. In the optimal control problem, both the initial and the desired terminal state are prescribed. We assume that the system is exactly controllable in an abstract sense if the time horizon is long enough. We show that that the corresponding optimal control problems on the time intervals $$[0, \, T]$$ [ 0 , T ] give rise to a turnpike structure in the sense that for natural numbers n if T is sufficiently large, the contribution of the objective function from subintervals of [0, T] of the form $$\begin{aligned} {[}t - t/2^n,\; t + (T-t)/2^n] \end{aligned}$$ [ t - t / 2 n , t + ( T - t ) / 2 n ] is of the order $$1/\min \{t^n, (T-t)^n\}$$ 1 / min { t n , ( T - t ) n } . We also show that a similar result holds for $$\epsilon $$ ϵ -optimal solutions of the optimal control problems if $$\epsilon >0$$ ϵ > 0 is chosen sufficiently small. At the end of the paper we present both systems that are governed by ordinary differential equations and systems governed by partial differential equations where the results can be applied.

We consider an Sp(4,Z) invariant expression involving two factors of the Kawazumi--Zhang (KZ) invariant each of which is a modular graph with one link, and four derivatives on the moduli space of genus two Riemann surfaces. Manipulating it, we show that the integral over moduli space of a linear combination of a modular graph with two links and the square of the KZ invariant reduces to a boundary integral. We also consider an Sp(4,Z) invariant expression involving three factors of the KZ invariant and six derivatives on moduli space, from which we deduce that the integral over moduli space of a modular graph with three links reduces to a boundary integral. In both cases, the boundary term is completely determined by the KZ invariant. We show that both the integrals vanish., Comment: 27 pages, LaTeX; 4 figures

We first prove an identity involving symmetric polynomials. This identity leads us into exploring the geometry of Lagrangian Grassmannians. As an insight applications, we obtain a formula for the integral over the Lagrangian Grassmannian of a characteristic class of the tautological sub-bundle. Moreover, a relation to that over the ordinary Grassmannian and its application to the degree formula for the Lagrangian Grassmannian are given. Finally, we present further applications to the computation of Schubert structure constants and three-point, degree 1, genus 0 Gromov–Witten invariants of the Lagrangian Grassmannian. Some examples together with explicit computations are presented.

Tolerance has been the most important moral characteristic and social value of a person since the inception of human society. Tolerance, which is defined as a spiritual phenomenon, manifests itself in various forms in all areas of life and determines the nature of communication in human behavior. Among them we can show everything, from the most primitive ancient forms of consciousness to forms of artistic and rational thinking. The essence of each of these forms of self-expression is mutual understanding, understanding and empathy. As human knowledge deepened, thoughts and ideas around this problem expanded. The history of the development of tolerance shows that today this moral and legal category is an integral part of the modern way of life.

We present both necessary and sufficient conditions for a convex closed shape such that for every convex function the average integral over the shape does not exceed the average integral over its boundary. It is proved that this inequality holds for n-dimensional parallelotopes, n-dimensional balls, and convex polytopes having the inscribed sphere (tangent to all its facets) with the centre in the centre of mass of its boundary.

The numerical solutions of boundary integral equations by the Boundary Element Method (BEM) have been applied in several areas of computational engineering and science such as elasticity and fracture mechanics. The BEM formulations often require the evaluation of complex singular and hypersingular integrals. Therefore, BEM requires special integration schemes for singular elements. The Subtraction Singularity Technique (SST) is a general procedure for evaluating such integrals, which allows for the integral over high-order curved boundary elements. The SST regularises these kernels through the Taylor expansion of the integral kernels around the source point. This study presents the expressions from Taylor expansions, which are required for the regularization of singular and hypersingular boundary integral equations of plane linear anisotropic elasticity. These expressions have been implemented into an academic BEM code, which enables the integration over high-order straight and curved boundary elements. The proposed scheme leads to excellent performance. The results obtained by the proposed scheme are in excellent agreement with reference responses available in the literature.

In the recent paper Bürgisser and Lerario (Journal für die reine und angewandte Mathematik (Crelles J), 2016) introduced a geometric framework for a probabilistic study of real Schubert Problems. They denoted by $$\delta _{k,n}$$ δ k , n the average number of projective k-planes in $${\mathbb {R}}\mathrm {P}^n$$ R P n that intersect $$(k+1)(n-k)$$ ( k + 1 ) ( n - k ) many random, independent and uniformly distributed linear projective subspaces of dimension $$n-k-1$$ n - k - 1 . They called $$\delta _{k,n}$$ δ k , n the expected degree of the real Grassmannian $${\mathbb {G}}(k,n)$$ G ( k , n ) and, in the case $$k=1$$ k = 1 , they proved that: $$\begin{aligned} \delta _{1,n}= \frac{8}{3\pi ^{5/2}} \cdot \left( \frac{\pi ^2}{4}\right) ^n \cdot n^{-1/2} \left( 1+{\mathcal {O}}\left( n^{-1}\right) \right) . \end{aligned}$$ δ 1 , n = 8 3 π 5 / 2 · π 2 4 n · n - 1 / 2 1 + O n - 1 . Here we generalize this result and prove that for every fixed integer $$k>0$$ k > 0 and as $$n\rightarrow \infty $$ n → ∞ , we have $$\begin{aligned} \delta _{k,n}=a_k \cdot \left( b_k\right) ^n\cdot n^{-\frac{k(k+1)}{4}}\left( 1+{\mathcal {O}}(n^{-1})\right) \end{aligned}$$ δ k , n = a k · b k n · n - k ( k + 1 ) 4 1 + O ( n - 1 ) where $$a_k$$ a k and $$b_k$$ b k are some (explicit) constants, and $$a_k$$ a k involves an interesting integral over the space of polynomials that have all real roots. For instance: $$\begin{aligned} \delta _{2,n}= \frac{9\sqrt{3}}{2048\sqrt{2\pi }} \cdot 8^n \cdot n^{-3/2} \left( 1+{\mathcal {O}}\left( n^{-1}\right) \right) . \end{aligned}$$ δ 2 , n = 9 3 2048 2 π · 8 n · n - 3 / 2 1 + O n - 1 . Moreover we prove that these numbers belong to the ring of periods intoduced by Kontsevich and Zagier and give an explicit formula for $$\delta _{1,n}$$ δ 1 , n involving a one-dimensional integral of certain combination of Elliptic functions.

For boundary element analysis of the orthotropic potential problems in thin structures, the higher order elements are expected to discretize the boundary. However, the use of the higher order elements leads to more complex forms of the integrands in boundary integral equations. The resulting nearly singular integrals on the higher order elements are difficult to be evaluated when the source point is very close to the integral element. In this paper, a semi-analytic algorithm is presented to evaluate the nearly singular integrals on the quadratic elements in two dimensional (2-D) orthotropic potential problems. By constructing the approximate singular integral kernels, the nearly singular integrals through subtraction technique are transformed into the sum of regular parts and singular parts. Then, the former are calculated by the conventional Gaussian quadrature and the latter are calculated by the analytical integral formulas. Numerical examples demonstrate that the present semi-analytic algorithm is efficient and accurate to calculate the nearly singular integrals on the quadratic elements. Especially, the BEM with the present semi-analytic algorithm is successfully applied to analyzing 2-D orthotropic potential problems in very thin structures.

The Cattaneo equations with Caputo–Fabrizio fractional derivative are investigated. A compact finite difference scheme of Crank–Nicolson type is presented and analyzed, which is proved to have temporal accuracy of second order and spatial accuracy of fourth order. Since this derivative is defined with an integral over the whole passed time, conventional direct solvers generally take computational complexity of OMN2 and require memory of OMN, with M and N the number of space steps and time steps, respectively. We develop a fast evaluation procedure for the Caputo–Fabrizio fractional derivative, by which the computational cost is reduced to OMN operations; meanwhile, only OM memory is required. In the end, several numerical experiments are carried out to verify the theoretical results and show the applicability of the fast compact difference procedure.

We consider the $S^3\times S^1$ superconformal index $\mathcal{I}(\tau)$ of 4d $\mathcal{N}=1$ gauge theories. The Hamiltonian index is defined in a standard manner as the Witten index with a chemical potential $\tau$ coupled to a combination of angular momenta on $S^3$ and the $U(1)$ R-charge. We develop the all-order asymptotic expansion of the index as $q = e^{2 \pi i \tau}$ approaches a root of unity, i.e. as $\widetilde \tau \equiv m \tau + n \to 0$, with $m,n$ relatively prime integers. The asymptotic expansion of $\log\mathcal{I}(\tau)$ has terms of the form $\widetilde \tau^k$, $k = -2, -1, 0, 1$. We determine the coefficients of the $k=-2,-1,1$ terms from the gauge theory data, and provide evidence that the $k=0$ term is determined by the Chern-Simons partition function on $S^3/\mathbb{Z}_m$. We explain these findings from the point of view of the 3d theory obtained by reducing the 4d gauge theory on the $S^1$. The supersymmetric functional integral of the 3d theory takes the form of a matrix integral over the dynamical 3d fields, with an effective action given by supersymmetrized Chern-Simons couplings of background and dynamical gauge fields. The singular terms in the $\widetilde \tau \to 0$ expansion (dictating the growth of the 4d index) are governed by the background Chern-Simons couplings. The constant term has a background piece as well as a piece given by the localized functional integral over the dynamical 3d gauge multiplet. The linear term arises from the supersymmetric Casimir energy factor needed to go between the functional integral and the Hamiltonian index., Comment: v3: minor corrections and clarifications added

We determine the correlation energy of BN, SiO$_2$ and ice polymorphs employing a recently developed RPAx (random phase approximation with exchange) approach. The RPAx provides larger and more accurate polarizabilities as compared to the RPA, and captures effects of anisotropy. In turn, the correlation energy, defined as an integral over the density-density response function, gives improved binding energies without the need for error cancellation. Here, we demonstrate that these features are crucial for predicting the relative energies between low- and high-pressure polymorphs of different coordination number as, e.g., between $\alpha$-quartz and stishovite in SiO$_2$, or layered and cubic BN. Furthermore, a reliable (H$_2$O)$_2$ potential energy surface is obtained, necessary for describing the various phases of ice. The RPAx gives results comparable to other high-level methods such as coupled cluster and quantum Monte Carlo, also in cases where the RPA breaks down. Although higher computational cost than RPA we observe a faster convergence with respect to the number of eigenvalues in the response function., Comment: 13 pages + Appendices, 9 figures

In this study we use a boundary integral element-based numerical technique to solve the generalized Burger-Fisher equation. The essential feature of this method is the fundamental integral representation of the solution inside the problem domain by means of both the boundary and domain values. The occurrences of domain integrals within the problem arising from nonlinearity as well as the temporal derivative are not avoided or transferred to the boundary. However, unlike the classical boundary element approach, they are resolved within a finite-element-type discrete domain. The utility and correctness of this formulation are proved by comparing the results obtained herein with closed form solutions.

The expansion of a modular graph function on a torus of modulus $\tau$ near the cusp is given by a Laurent polynomial in $y= \pi \Im (\tau)$ with coefficients that are rational multiples of single-valued multiple zeta-values, apart from the leading term whose coefficient is rational and exponentially suppressed terms. We prove that the coefficients of the non-leading terms in the Laurent polynomial of the modular graph function $D_N(\tau)$ associated with a melon graph is free of irreducible multiple zeta-values and can be written as a polynomial in odd zeta-values with rational coefficients for arbitrary $N \geq 0$. The proof proceeds by expressing a generating function for $D_N(\tau)$ in terms of an integral over the Virasoro-Shapiro closed-string tree amplitude.

The singularity cancelation is usually of great importance in the calculation of nearly singular integral that is involved in the boundary element analysis of thin structures and in the extended finite element method for crack problems. The well-known distance transformation can be performed for various kinds of singular kernels to compute the near singular integral. However, in the case of that the source point is located nearby the boundary of the integral element, the result is usually very sensitive to the shape of the sub-triangles. A Distance-Sinh combined transformation method based on the generalized Duffy normalization is proposed in this paper to improve the accuracy of the nearly singular integral calculation. Furthermore, a two-level transformation method is proposed to cancel the singularity of the integrand. In first level of the method, the generalized Duffy transformation is implemented. In the transformed integral space (u, v), the singularity of the integrand along u-direction is eliminated by the distance transformation. The singularity of the integrand along the other direction is canceled by the Sinh transformation. Numerical examples are presented to verify the proposed method.

Let A ⊆ B be an extension of commutative rings with identity and T ( A ) (resp., T ( B ) ) be the total quotient ring of A (resp., B). We say that A is a B-Noetherian ring if every ideal I of A with I B = B is finitely generated, and the extension A ⊆ B is called a Noetherian extension if A is B-Noetherian. In this paper, among other things, we introduce the ring A g ( B ) , the global transform of A in B. We then show that if x ∈ A is B-regular (i.e., x B = B ) and R is a ring between A and A g ( B ) , then R / x R is a Noetherian A-module and every ideal of R containing x is finitely generated. We also study some conditions on A and B under which if A is B-Noetherian with dim B A = 1 , then any ring between A and B is B-Noetherian. This is the Krull-Akizuki theorem when A is an integral domain and B = T ( A ) . Finally, we prove that if T ( B ) is integral over T ( A ) and A is integrally closed in B, then every ideal I of A with I B = B is t-invertible, i.e., ( I I − 1 ) t = A .

We deal with an inverse source problem in a time-fractional wave equation. The time-dependent source term is reconstructed using the additional non-invasive measurement in the form of integral over a part of the boundary. We look for the solution and the source term obeying the variational formulation and the equation obtained from applying the measurement on the equation. Using the Rothe method the existence of the solution is proven. The uniqueness is shown in appropriate spaces and some numerical experiments are presented.

Even for holographic theories that obey boundary causality, the full bulk Lorentzian path integral includes metrics that violate this condition. This leads to the following puzzle: The commutator of two field theory operators at spacelike-separated points on the boundary must vanish. However, if these points are causally related in a bulk metric, then the bulk calculation of the commutator will be nonzero. It would appear that the integral over all metrics of this commutator must vanish exactly for holography to hold. This is puzzling since it must also be true if the commutator is multiplied by any other operator. Upon a careful treatment of boundary conditions in holography, we show how the bulk path integral leads to a natural resolution of this puzzle., Comment: 14 pages, 1 figure; v2: matches published version

The character integral representation of one loop partition functions is useful to establish the relation between partition functions of conformal fields on Weyl equivalent spaces. The Euclidean space $S^a\times AdS_b$ can be mapped to $S^{a+b}$ provided $S^a$ and $AdS_b$ are of the same radius. As an example, to begin with, we show that the partition function in the character integral representation of conformally coupled free scalars and fermions are identical on $S^a\times AdS_b$ and $S^{a+b}$. We then demonstrate that the partition function of higher derivative conformal scalars and fermions are also the same on hyperbolic cylinders and branched spheres. The partition function of the four-derivative conformal vector gauge field on the branched sphere in $d=6$ dimension can be expressed as an integral over `naive' bulk and `naive' edge characters. However, the partition function of the conformal vector gauge field on $S^1_q\times AdS_5$ contains only the `naive' bulk part of the partition function. This follows the same pattern which was observed for the partition of conformal $p$-form fields on hyperbolic cylinders. We use the partition function of higher derivative conformal fields on hyperbolic cylinders to obtain a linear relationship between the Hofman-Maldacena variables which enables us to show that these theories are non-unitary., 44 pages, 3 tables and 7 figures

This research provides an insight on the performances of machine learning (ML)-based algorithms for the estimation of the energy consumption in metal forming processes and is applied to the radial-axial ring rolling process. To define the mutual influence between ring geometry, process settings, and ring rolling mill geometries with the resulting energy consumption, measured in terms of the force integral over the processing time (FIOT), FEM simulations have been implemented in the commercial SW Simufact Forming 15. A total of 380 finite element simulations with rings ranging from 650 mm &lt, DF &lt, 2000 mm have been implemented and constitute the bulk of the training and validation datasets. Both finite element simulation settings (input), as well as the FI (output), have been utilized for the training of eight machine learning models, implemented with Python scripts. The results allow defining that the Gradient Boosting (GB) method is the most reliable for the FIOT prediction in forming processes, being its maximum and average errors equal to 9.03% and 3.18%, respectively. The trained ML models have been also applied to own and literature experimental cases, showing a maximum and average error equal to 8.00% and 5.70%, respectively, thus proving once again its reliability.

We calculate the four-point function of 1/2-BPS (Bogomolnyi–Prasad–Sommerfield) determinant operators in N=4 SYM (super Yang-Mills) at next-to-leading order at weak coupling. We use two complementary methods recently developed for a class of determinant three-point functions: one is based on Feynman diagrams and it extracts perturbative data at finite N, while the other one expresses a generic correlator of determinants as the zero-dimensional integral over an auxiliary matrix field. We generalize the latter approach to calculate one-loop corrections and we solve the four-point function in a semiclassical approach at large N. The results allow us to comment on the order of the phase transition that the four-point function is expected to exhibit in an exact integrability-based description.

The two linear relations between operators shown in eq. (3.29) were missing an integral over the momentum fraction u on the right-hand side. In the one-particle anomalous dimensions in eq. (5.7) two fractions were mistyped.

We consider the Hartle-Hawking wavefunction of the universe defined as a Euclidean path integral that satisfies the "no-boundary proposal." We focus on the simplest minisuperspace model that comprises a single scale factor degree of freedom and a positive cosmological constant. The model can be seen as a non-linear $\sigma$-model with a line-segment base. We reduce the path integral over the lapse function to an integral over the proper length of the base and use diffeomorphism-invariant measures for the ghosts and the scale factor. As a result, the gauge-fixed path integral is independent of the gauge. However, we point out that all field redefinitions of the scale factor degree of freedom yield different choices of gauge-invariant path-integral measures. For each prescription, we compute the wavefunction at the semi-classical level and find a different result. We resolve in each case the ambiguity in the form of the Wheeler-DeWitt equation at this level of approximation. By imposing that the Hamiltonians associated with these possibly distinct quantum theories are Hermitian, we determine the inner products of the corresponding Hilbert spaces and find that they lead to a universal norm, at least semi-classically. Quantum predictions are thus independent of the prescription at this level of approximation. Finally, all wavefunctions of the Hilbert spaces of the minisuperspace model we consider turn out to be non-normalizable, including the no-boundary states., Comment: 1+56 pages, 1 figure

We present a theory of viscoelasticity of amorphous media, which takes into account the effects of confinement along one of three spatial dimensions. The framework is based on the nonaffine extension of lattice dynamics to amorphous systems, or nonaffine response theory. The size effects due to the confinement are taken into account via the nonaffine part of the shear storage modulus ${G}^{\ensuremath{'}}$. The nonaffine contribution is written as a sum over modes in $k$-space. With a rigorous argument based on the analysis of the $k$-space integral over modes, it is shown that the confinement size $L$ in one spatial dimension, e.g., the $z$ axis, leads to a infrared cutoff for the modes contributing to the nonaffine (softening) correction to the modulus that scales as ${L}^{\ensuremath{-}3}$. Corrections for finite sample size $D$ in the two perpendicular dimensions scale as $\ensuremath{\sim}{(L/D)}^{4}$, and are negligible for $L\ensuremath{\ll}D$. For liquids it is predicted that ${G}^{\ensuremath{'}}\ensuremath{\sim}{L}^{\ensuremath{-}3}$ is in agreement with a previous more approximate analysis, whereas for amorphous materials ${G}^{\ensuremath{'}}\ensuremath{\sim}{G}_{\text{bulk}}^{\ensuremath{'}}+\ensuremath{\beta}{L}^{\ensuremath{-}3}$. For the case of liquids, four different experimental systems are shown to be very well described by the ${L}^{\ensuremath{-}3}$ law. The theory can also explain previous simulation data of confined jammed granular packings.

The Coulomb Branch Formula conjecturally expresses the refined Witten index for $N=4$ Quiver Quantum Mechanics as a sum over multi-centered collinear black hole solutions, weighted by so-called `single-centered' or `pure-Higgs' indices, and suitably modified when the quiver has oriented cycles. On the other hand, localization expresses the same index as an integral over the complexified Cartan torus and auxiliary fields, which by Stokes' theorem leads to the famous Jeffrey-Kirwan residue formula. Here, by evaluating the same integral using steepest descent methods, we show the index is in fact given by a sum over deformed multi-centered collinear solutions, which encompasses both regular and scaling collinear solutions. As a result, we confirm the Coulomb Branch Formula for Abelian quivers in the presence of oriented cycles, and identify the origin of the pure-Higgs and minimal modification terms as coming from collinear scaling solutions. For cyclic Abelian quivers, we observe that part of the scaling contributions reproduce the stacky invariants for trivial stability, a mathematically well-defined notion whose physics significance had remained obscure., 45 pages; v2: added subsection 3.4.2 on Abelian quivers with multiple oriented cycles, plus various clarifications and several references. Converted to SciPost style

It is well known that the low energy expansion of tree-level superstring scattering amplitudes satisfies a suitably defined version of maximum transcendentality. In this paper it is argued that there is a natural extension of this definition that applies to the genus-one four-graviton Type II superstring amplitude to all orders in the low-energy expansion. To obtain this result, the integral over the genus-one moduli space is partitioned into a region ${\cal M}_R$ surrounding the cusp and its complement ${\cal M}_L$, and an exact expression is obtained for the contribution to the amplitude from ${\cal M}_R$. The low-energy expansion of the ${\cal M}_R$ contribution is proven to be free of irreducible multiple zeta-values to all orders. The contribution to the amplitude from ${\cal M}_L$ is computed in terms of modular graph functions up to order $D^{12} {\cal R}^4$ in the low-energy expansion, and general arguments are used beyond this order to conjecture the transcendentality properties of the ${\cal M}_L$ contributions. Maximal transcendentality of the full amplitude holds provided we assign a non-zero weight to certain harmonic sums, an assumption which is familiar from transcendentality assignments in quantum field theory amplitudes., Comment: 65 pages, 4 figures; typos corrected, reference added, minor edits in version 2; factor of 4 corrected in theorem 4.1 in version 3

Angelani et al. (1) suggest an alternative derivation of the relation between low-frequency shear modulus G ′ and confinement length L derived in ref. 2. They seem to imply that an integral over plane waves in three-dimensional (3D) isotropic media can be decoupled into a separate integral over k z (z component of wavevector k, the confined dimension), times an independent 2D integral in the unconfined ( k x , k y ) plane. This would then change the final integration in equation 9 of ref. 2 from G ′ = G ∞ − α ∫ 1 / L k D k 2 d k to G ′ = G ∞ − γ ∫ 1 / L k D d k , where α and γ are constants, and the scaling of G ′ with L … [↵][1]1To whom correspondence may be addressed. Email: az302{at}cam.ac.uk. [1]: #xref-corresp-1-1

We characterize, using normal forms of quasi-homogeneous expansions, the analytic vector fields at nilpotent singular point having an algebraic first integral over the ring C [[ x, y ]] . As a consequence, we provide a link between the algebraic integrability problem and the existence of a formal inverse integrat- ing factor which is null at the singular point., This work has been partially supported by Ministerio de Ciencia, Innovación y Universidades, Spain (project PGC2018-096265-BI00) and by Consejería de Economía, Innovación, Ciencia y Empleo de la Junta de Andalucía, Spain (projects FQM-276, UHU-1260150 and P12-FQM-1658 Funding for open access charge: Universidad de Huelva / CBUA.

We characterize non-Hermitian band structures by symmetry indicator topological invariants. Enabled by crystalline inversion symmetry, these indicators allow us to short-cut the calculation of conventional non-Hermitian topological invariants. In particular, we express the three-dimensional winding number of point-gapped non-Hermitian systems, which is defined as an integral over the whole Brillouin zone, in terms of symmetry eigenvalues at high-symmetry momenta. Furthermore, for time-reversal symmetric non-Hermitian topological insulators, we find that symmetry indicators characterize the associated Chern-Simons form, whose evaluation usually requires a computationally expensive choice of smooth gauge. In each case, we discuss the non-Hermitian surface states associated with nontrivial symmetry indicators., Comment: 6 pages, 1 figure, supplement included

A new angular quadrature type termed Modified Simpson Trapezoidal (MST) is developed based on the conventional Simpson’s 1/3 rule where the angular pattern over polar levels has a trapezoid shape. An adaptive coefficient correction scheme is developed to enable our new quadrature to integrate the angular flux over subintervals separated by the interior jump irregularities. A two-dimensional test problem is employed to verify the angular discretization error in the uncollided SN scalar flux computed with our new quadrature sets, as well as conventional angular quadrature types. Numerical results show that the MST quadrature error in the point-wise scalar flux converges with second order against increasing number of discrete angles, while the error obtained with other conventional quadrature types converges slower than first order depending on the regularity of the exact point-wise uncollided angular flux. In order to reduce the number of discrete points needed, a variant of the MST quadrature, namely MSTP30, is developed by using the Quadruple Range [1] polar quadrature with fixed 30 polar angles and applying the MST quadrature to the azimuthal dependence in each polar level. The angular discretization error in the point-wise SN scalar flux obtained with MSTP30 sets converges with fourth order because the polar discretization error is sufficiently reduced that MSTP30 behaves like a one-dimensional quadrature. Furthermore, because MSTP30 computes the integral over subintervals that keep the true solution’s irregularity at the boundaries, this fourth order convergence rate is unaffected by such inevitable irregularities.

In this paper, we consider an inverse problem for three dimensional viscoelastic fluid flow equations, which arises from the motion of Kelvin-Voigt fluids in bounded domains (a hyperbolic type problem). This inverse problem aims to reconstruct the velocity and kernel of the memory term simultaneously, from the measurement described as the integral over determination condition. By using the contraction mapping principle in an appropriate space, a local in time existence and uniqueness result for the inverse problem of Kelvin-Voigt fluids are obtained. Furthermore, using similar arguments, a global in time existence and uniqueness results for an inverse problem of Oseen type equations are also achieved.

We provide an analytical proof of universality for bound states in one-dimensional systems of two and three particles, valid for short-range interactions with negative or vanishing integral over space. The proof is performed in the limit of weak pair-interactions and covers both binding energies and wave functions. Moreover, in this limit the results are formally shown to converge to the respective ones found in the case of the zero-range contact interaction.

One of the most suitable methods for modeling fully dynamic earthquake cycle simulations is the spectral boundary integral element method (sBIEM), which takes advantage of the fast Fourier transform (FFT) to make a complex numerical dynamic rupture tractable. However, this method has the serious drawback of requiring a flat fault geometry due to the FFT approach. Here, we present an analytical formulation that extends the sBIEM to a mildly nonplanar fault. We start from a regularized boundary element method and apply a small-slope approximation of the fault geometry. Making this assumption, it is possible to show that the main effect of nonplanar fault geometry is to change the normal traction along the fault, which is controlled by the local curvature along the fault. We then convert this space–time boundary integral equation of the normal traction into a spectral-time formulation and incorporate this change in normal traction into the existing sBIEM methodology. This approach allows us to model fully dynamic seismic cycle simulations on nonplanar faults in a particularly efficient way. We then test this method against a regular BIEM for both rough-fault and seamount-fault geometries and demonstrate that this sBIEM maintains the scaling between the fault geometry and slip distribution.

In 1944 Atle Selberg published a remarkable n-dimensional analogue of Euler's beta integral which now bears his name. The Selberg integral has come to be regarded as one of the most important hypergeometric integrals, a reputation which is upheld by its uses in fields such as random matrix theory, analytic number theory, conformal field theory and enumerative and algebraic combinatorics. In their verification of the AGT conjecture for SU(2), Alba, Fateev, Litvinov and Tarnopolsky (AFLT) discovered a new generalisation of the Selberg integral over a pair of Jack polynomials. The AFLT integral unifies the well-known Kadell and Hua–Kadell integrals. The purpose of this thesis is to investigate generalisations of the AFLT integral in several directions, all based on symmetric function theory. Using new Cauchy-type identities for Macdonald polynomials we present two An Selberg integrals over a pair of Jack polynomials; one directly generalising the AFLT integral, and one generalising the A2 Selberg integral of Warnaar. Following Matsuo and Zhang, we then consider An Selberg integrals with n+1 symmetric functions in the integrand. Here our results are restricted to the Schur case, where we use several new integral formulas for complex Schur functions to evaluate the An integral first considered by Matsuo and Zhang. To conclude, we discuss other recent developments including the elliptic AFLT integral, an AFLT integral for Macdonald polynomials, an Askey–Habseiger–Kadell-type q-AFLT integral, and several open problems.

An integral is defined on the plane that includes the Henstock-Kurzweil and Lebesgue integrals (with respect to Lebesgue measure). A space of primitives is taken as the set of continuous real-valued functions \(F(x,y)\) defined on the extended real plane \([-\infty,\infty]^2\) that vanish when \(x\) or \(y\) is \(-\infty\). With usual pointwise operations this is a Banach space under the uniform norm. The integrable functions and distributions (generalised functions) are those that are the distributional derivative \(\partial^2/(\partial x\partial y)\) of this space of primitives. If \(f=\partial^2/(\partial x\partial y) F\) then the integral over interval \([a,b]\times [c,d] \subseteq[-\infty,\infty]^2\) is \(\int_a^b\int_c^d f=F(a,c)+F(b,d)-F(a,d)-F(b,c)\) and \(\int_{-\infty}^\infty \int_{-\infty}^\infty f=F(\infty,\infty)\). The definition then builds in the fundamental theorem of calculus. The Alexiewicz norm is \({\lVert f\rVert}={\lVert F\rVert}_\infty\) where \(F\) is the unique primitive of \(f\). The space of integrable distributions is then a separable Banach space isometrically isomorphic to the space of primitives. The space of integrable distributions is the completion of both \(L^1\) and the space of Henstock-Kurzweil integrable functions. The Banach lattice and Banach algebra structures of the continuous functions in \({\lVert \cdot\rVert}_\infty\) are also inherited by the integrable distributions. It is shown that the dual space is the functions of bounded Hardy-Krause variation. Various tools that make these integrals useful in applications are proved: integration by parts, Hölder’s inequality, second mean value theorem, Fubini’s theorem, a convergence theorem, change of variables, convolution. The changes necessary to define the integral in \({\mathbb R}^n\) are sketched out.

It is shown that if $R$ is a ring, $p$ a prime element of an integral domain $D\leq R$ with $\bigcap_{n=1}^\infty p^nD=0$ and $p\in U(R)$, then $R$ has a conch maximal subring (see \cite{faith}). We prove that either a ring $R$ has a conch maximal subring or $U(S)=S\cap U(R)$ for each subring $S$ of $R$ (i.e., each subring of $R$ is closed with respect to taking inverse, see \cite{invsub}). In particular, either $R$ has a conch maximal subring or $U(R)$ is integral over the prime subring of $R$. We observe that if $R$ is an integral domain with $|R|=2^{2^{\aleph_0}}$, then either $R$ has a maximal subring or $|Max(R)|=2^{\aleph_0}$, and in particular if in addition $dim(R)=1$, then $R$ has a maximal subring. If $R\subseteq T$ be an integral ring extension, $Q\in Spec(T)$, $P:=Q\cap R$, then we prove that whenever $R$ has a conch maximal subring $S$ with $(S:R)=P$, then $T$ has a conch maximal subring $V$ such that $(V:T)=Q$ and $V\cap R=S$. It is shown that if $K$ is an algebraically closed field which is not algebraic over its prime subring and $R$ is affine ring over $K$, then for each prime ideal $P$ of $R$ with $ht(P)\geq dim(R)-1$, there exists a maximal subring $S$ of $R$ with $(S:R)=P$. If $R$ is a normal affine integral domain over a field $K$, then we prove that $R$ is an integrally closed maximal subring of a ring $T$ if and only if $dim(R)=1$ and in particular in this case $(R:T)=0$.

The Cottingham formula expresses the leading contribution of the electromagnetic interaction to the proton-neutron mass difference as an integral over the forward Compton amplitude. Since quarks and gluons reggeize, the dispersive representation of this amplitude requires a subtraction. We assume that the asymptotic behaviour is dominated by Reggeon exchange. This leads to a sum rule that expresses the subtraction function in terms of measurable quantities. The evaluation of this sum rule leads to $m_{QED}^{p-n}=0.58\pm 0.16\,\mbox{MeV}$., 26 pages, 9 figures. Update, matches the version published in EPJC

A general procedure for obtaining frame-independent three-dimensional light-front coordinate-space wave functions is introduced. The third spatial coordinate $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{z}$ is the boost and Lorentz frame-independent coordinate conjugate to the light-front momentum coordinate $x=\frac{{k}^{+}}{{P}^{+}}$ which appears in the momentum-space light-front wave functions underlying generalized parton distributions, structure functions, distribution amplitudes, form factors, and other hadronic observables. These causal light-front coordinate-space wave functions are used to derive a general expression for the quark distribution function of hadrons as an integral over the frame-independent longitudinal distance (the Ioffe time) between virtual-photon absorption and emission appearing in the forward virtual photon-hadron Compton scattering amplitude. Specific examples using models derived from light-front holographic QCD show that the spatial extent of the proton eigenfunction in the longitudinal direction can have a very large extent in $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{z}$.

We recast the joint $$ J\overline{T} $$ J T ¯ , $$ T\overline{J} $$ T J ¯ and $$ T\overline{T} $$ T T ¯ deformations as coupling the original theory to a mixture of topological gravity and gauge theory. This geometrizes the general flow triggered by irrelevant deformations built out of conserved currents and the stress-energy tensor, by means of a path integral kernel. The partition function of the deformed theory satisfies a diffusion-like flow equation similar to that found in the pure $$ T\overline{T} $$ T T ¯ case. Our proposal passes two stringent tests. Firstly, we recover the classical deformed actions from the kernel, reproducing the known expressions for the free boson and fermion. Secondly, we explicitly compute the torus path integral along the flow and show it localizes to a finite-dimensional, one-loop exact integral over base space torus moduli. The dressed energy levels so obtained match exactly onto those previously reported in the literature.

The isogeometric boundary element method (IGABEM) has great potential for the simulation of elasticity problems because of its exact geometric representation and good approximation properties. These advantages can be extended to thin structures, including coatings, but the development of an accurate and efficient method to deal with the large number of nearly singular integrals existing in the IGABEM presents a great challenge for very thin sections. In this paper, we propose a new sinh + scheme for weakly, strongly and hyper near-singular integrals arising in 3D IGABEM for thermoelastic problems, based on the sinh transformation method and adaptive integral method. The presented scheme is efficient, since it combines the advantages of both methods: (1) when the thickness δ of coatings/thin structures is moderately small, an accurate and efficient integral result will be obtained by the adaptive integral method; (2) when δ is very small, the nearly singular integrals are computed by the sinh + scheme efficiently. With the introduction of NURBS in IGABEM, truncation errors arising in the Taylor expansion cannot be ignored. Based on the values of these errors, the computed knot spans are further divided into several sub-knot spans and different methods will be used to evaluate the integral over each sub-knot span in the new scheme. In addition, based on the analytical extension of the NURBS surface, an adaptation of the sinh transformation method is proposed which can evaluate the near-singular integrals accurately for cases in which the projection point lies outside of the considered knot span. Several numerical examples are presented to validate the accuracy and efficiency of the 3D IGABEM based on the sinh + scheme in the analysis of thermoelastic problems.

The connection is established between two theories that have developed independently with the aim to describe quantum mechanics as a stochastic process, namely stochastic quantum mechanics (sqm) and stochastic electrodynamics (sed). Important commonalities and complementarities between the two theories are identified, notwithstanding their dissimilar origins and approaches. Further, the dynamical equation of sqm is completed with the radiation terms that are an integral element in sed. The central problem of the transition to the quantum dynamics is addressed, pointing to the key role of diffusion in the emergence of quantization.