Search

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

This article explores the mathematical description of anomalous diffusion, driven not by thermal fluctuations but by internal stresses. A continuous time random walk framework is outlined in which the waiting times between displacements (jumps), generated by the dynamics of internal stresses, are described by the generalized $\mathrm{\ensuremath{\Gamma}}$ distribution. The associated generalized diffusion equation is then identified. The solution to this equation is obtained as an integral over an infinite series of Fox $H$ functions. The probability density function is identified as initially non-Gaussian, while at longer timescales Gaussianity is recovered. Likewise, the second moment displays a transient nature, shifting between subdiffusive and diffusive character. The potential application of this mathematical description to the quaking observed in several soft-matter systems is discussed briefly.

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 study matrices over arbitrary rings that are integral over the base ring. For any ring [Formula: see text], let [Formula: see text] with [Formula: see text]. We prove that for any pairwise commuting elements [Formula: see text] if [Formula: see text], then [Formula: see text]. As a corollary, it follows that for [Formula: see text], [Formula: see text] commutative ring, if [Formula: see text] are pairwise commuting matrices such that [Formula: see text], then [Formula: see text] where [Formula: see text]. This result, which is a generalization of the Cayley–Hamilton Theorem, was proved by Phillips in 1919. For a positive integer [Formula: see text], we prove that if every matrix in [Formula: see text] satisfies a monic polynomial of degree [Formula: see text] over [Formula: see text], then [Formula: see text] is commutative. On the other hand, every diagonal matrix in [Formula: see text], [Formula: see text], satisfies a monic polynomial of degree [Formula: see text] over [Formula: see text] precisely when [Formula: see text] is a left duo ring. We prove that if every diagonal matrix in [Formula: see text], [Formula: see text], is [Formula: see text]-integral, then [Formula: see text] is Dedekind finite.

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.

Energy is indeed an integral element for daily life. Energy has become a highly vital utility for every country and also its availability will impact the entire output of financial resources in a dynamic manner.

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.

For a smooth mapping \(f : \overline D \subset {\mathbb R}^n \to {\mathbb R}^n\) with 0∉f(∂D), D open, bounded with smooth boundary ∂D, the Kronecker index of f on ∂D is defined as the integral over ∂D of some (n − 1)-differential form depending upon f and its partial derivatives. For n = 2, it is the Cauchy index or winding number of f on ∂D, the algebraic number of turns of f around the closed curve ∂D. The Gauss linking number of two curves is a special case of the Kronecker index in \(\mathbb R^3\). Various expressions of the Kronecker index are given, as well as applications to the Ginzburg-Landau theory of liquid crystals. Writing the Kronecker index as an integral over \(\overline D\) leads in a natural way to the more general concept of the Brouwer degree for a continuous mappings \(f : \overline D \to {\mathbb R}^n\) without smoothness assumption upon ∂D. Its fundamental properties are established, including an axiomatic characterization and its localization to a neighborhood of an isolated zero, the Brouwer index. Applications are given to generalized implicit function theorems. An easy extension is made to mappings \(f : \overline D \subset X \to Y\) between oriented normed spaces X and Y of the same finite dimension, and some results on its computation are given, including reductions to situations of smaller dimension. Borsuk’s fruitful topological concept of retract leads to the obtention of the Brouwer fixed point theorem for continuous self-mappings of various classes of sets, and to the definition of an index of fixed points for continuous mappings defined in retracts of \({\mathbb R}^n\) like convex sets, wedges or cones in \({\mathbb R}^n\). The role of dimension is illustrated by the hairy ball theorem and the Brouwer degree is applied to the linking of curves and its use in various minimax theorems of critical point theory.

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.

The Partial Element Equivalent Circuit (PEEC) method is widely used to model a large variety of electrical systems ranging from interconnects to antennas and packaging. A key aspect in the development of the method is the computation of interaction integrals, namely partial inductances and coefficients of potential. In this work, a novel semi-analytical approach to the computation of the self-interaction integral over rectangular domains is proposed for the zero thickness case as it occurs in the case of the coefficients of potential and partial inductances of very thin conductors and in the surface integral equation based formulation. The proposed semi-analytical formula is verified by comparison with more standard integration schemes.

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 control method of the opto-electronic tracking servo systems is a subject that scholars of various countries compete to study. This article introduces the integral element based on the classic double closed-loop control method to improve the system order and steady-state accuracy. The fuzzy controller is used to dynamically adjust the gain of the integral element to construct a fuzzy order-II control system, which solves the problems of large and unstable oscillations of high-order systems. And the effectiveness of the control method is proved by the quantitative simulation analysis.

The purpose of this work is to obtain the degree of the exceptional component of the space of holomorphic foliations of degree two and codimension one in $${\mathbb {P}}^3$$. This component is the closure of the orbit of the foliation defined by the differential form $$\begin{aligned} \omega =(3fdg-2gdf)/x_0,\quad \text{ where } f=x_0^2x_3-x_0x_1x_2+\dfrac{x_1^3}{3},\ g=x_0x_2-\dfrac{x_1^2}{2} \end{aligned}$$under the natural action of the group of automorphisms of $${\mathbb {P}}^3$$. Our first task is to unravel a geometric characterization of the pair g, f. This leads us to the construction of a parameter space as an explicit fiber bundle over the variety of complete flags. Using tools from equivariant intersection theory, especially Bott’s formula, the degree is expressed as an integral over our parameter space.

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.

The Cauchy type integral over curve $$\Gamma $$ is traditional tool for solving of boundary-value problems of complex analysis. But it can diverge if length of the curve is infinite. We use Hadamard’s concept of finite part of integral for investigation of that situation.

The article deals with the trade unions as a grant to protect the rights and interests of civil servants, reveals the main tasks of trade unions. The activity of trade union organizations in the structure of the state body in Ukraine is analyzed. The legal mechanisms of asserting the violated rights of a civil servant by a trade union organization of a public body and the role of trade unions in protecting the rights of civil servants in developed countries are emphasized. The state at certain times gave the trade unions great powers to protect the rights and interests of workers, and then deprived the trade unions of these powers. In connection with this, various problems arose in regulating the activities of trade unions in the protection of individual and collective rights and interests in the protection of public servants. All this affected the legal status of trade unions. However, it should be noted that trade unions are in constant flux and this leads to improvements in the regulations governing their activities. However, it should be noted that today there are many problems in Ukraine regarding the exercise by the trade union organizations of their powers in the civil service. In particular, the legal status of trade unions in the civil service is not regulated enough, which, in turn, does not allow them to fully protect the legal rights and interests of civil servants. Considering the importance of trade unions in protecting labor rights and the socio-economic interests of workers, in developing democratic forms of citizen participation in managing economic and political processes, a democratic, legal, and social state, which is Ukraine, should support trade unions and take care of legislative consolidation. their authority. Trade unions at all levels should once again return to the consideration of their core functions and pay attention to those that will now be more conducive to the achievement of the main objective of the creation and activity of trade unions – the protection of social-labor rights and interests of trade union members. Today’s Ukraine needs strong unions. A strong union is a union that effectively protects the interests of its members, enjoys their trust and support, is able to organize, if necessary, collective action to protect the socio-economic rights and interests of employees, has sufficient organizational, financial, and human resources to fulfill its statutory tasks. Keywords: trade union organization, protection, rights, the role of trade unions, legal mechanisms.

We study problems of optimal boundary control with systems governed by linear hyperbolic partial differential equations. The objective function is quadratic and given by an integral over the finite...

In this paper, a multiscale contour steered region integral (MCSRI) method is proposed to classify highly similar shapes with flexible interior connection architectures. A component distance map (CDM) is developed to robustly characterize the flexible interior connection structure, shape of the exterior contour, and their inter-relationship in a shape image. A novel multiscale region transform (MReT) is proposed to perform region integral over different contour-steered strips at all possible scales to effectively integrate patch features, and thus enables a better description of the shape image in a coarse-to-fine manner. It is applied to solve a challenging problem of classifying cultivars from leaf images, which is a new attempt in both biology and computer vision research communities. A soybean cultivar leaf vein database (SoyCultivarVein), which is the first cultivar leaf vein database, is created and presented for performance evaluation. The experimental results demonstrate the superiority of the proposed method over the state-of-the-art methods in similar shape classification and the possibility of cultivar recognition via leaf pattern analysis, which may lead to a new research interest towards fine-level shape analysis on cultivar classification.

We derive some Euler type double integral representations for hypergeometric functions in two variables. In the first part of this paper we deal with Horn’s H2 function, in the second part with Olsson’s FP function. Our double integral representing the FP function is compared with the formula for the same integral representing an H2 function by M. Yoshida (Hiroshima Math. J. 10 (1980), 329–335) and M. Kita (Japan. J. Math. 18 (1992), 25–74). As specified by Kita, their integral is defined by a homological approach. We present a classical double integral version of Kita’s integral, with outer integral over a Pochhammer double loop, which we can evaluate as H2 just as Kita did for his integral. Then we show that shrinking of the double loop yields a sum of two double integrals for FP.

This paper presents a new bivariate dimension reduction method (BDRM) for statistical moments evaluation and structural reliability analysis with accuracy and efficiency. A high-order unscented transformation (HUT) is introduced to evaluate the two-dimensional integrals involved in BDRM, and the free parameter involved in HUT is suggested. In this regard, the proposed BDRM can be formulated accordingly for statistical moments assessment. Then, the performance function’s probability density function (PDF) is reconstructed by the shifted generalized lognormal distribution (SGLD) with the available statistical moments as constraints. Thus, the failure probability can be straightforwardly obtained by a simple integral over the PDF. The proposed method is verified by five numerical examples, including linear and non-linear, explicit and implicit performance functions. Besides, some other existing methods are also employed to demonstrate the advantages of the proposed method. It is found that the proposed method can keep the trade-off of accuracy and efficiency for structural reliability analysis.

Topological insulators in odd dimensions are characterized by topological numbers. We prove the well-known relation between the topological number given by the Chern character of the Berry curvature and the Chern-Simons level of the low energy effective action for a general class of Hamiltonians bilinear in the fermion with general U(1) gauge interactions including nonminimal couplings by an explicit calculation. A series of Ward-Takahashi identities are crucial to relate the Chern-Simons level to a winding number, which could then be directly reduced to Chern character of Berry curvature by carrying out the integral over the temporal momenta.

The tasks involving repeated integral occur from time to time in technical practice. This paper introduces the research of authors in the field of repeated integrals within the required class of functions. Authors focus on the definite integral over repeated integral and they develop a tool for its computation. It involves two principal steps, analytical and numerical step. In the analytical step, the definite integral over a repeated integral is decomposed into n integrals and then the Cauchy form is used for further rearrangement. Numerical step involves Gauss type integration slightly modified by the authors. Several examples illustrating the operation of both analytical and numerical steps of the method are provided in the paper.

This study presents an innovative concept of a compact integrated solar-thermoelectric module that can form part of the building envelope. The heating/cooling modes use the photovoltaic electrical current to power the heat pump. The experimental analysis was carried out and the results of coefficient of performance were in the range 0.5–1 and 2.6–5 for cooling and heating functions, respectively. The study demonstrates that thermoelectric cooler can effectively be used for heating, ventilation, and air conditioning applications by integrating with solar panels especially in cooling applications. The system is environmentally friendly and can contribute in the implementation of zero energy buildings concept. Practical application: In order to help address the challenge of climate change and associated environmental effects, there is continuous demand for new technologies and applications that can be readily integrated into day-to-day life as a means of reducing anthropogenic impact. Heating, ventilation, and air conditioning, as one of the largest energy consumers in buildings, is the focus of many researchers seeking to reduce building energy use and environmental impact. This article proposes using facades and windows that have an integrated modules of solar photovoltaic cells and thermoelectric devices that are able to work together to achieve heating and cooling effects as required by the building without requiring any external operational power.

Recently, it has been observed that topological ultrametricity of data can be expressed as an integral over a function which describes local ultrametricity. It was then observed empirically that this function begins as a sharply decreasing function, in order to increase again back to one. After providing a method for estimating the falling part of the local ultrametricity of data, empirical evidence is given for its logistic behaviour in relation to the number of connected components of the Vietoris-Rips graphs involved. The result is a functional dependence between that number and the number of maximal cliques. Further, it turns out that the logistic parameters depend linearly on the datasize. These observations are interpreted in terms of the Erdős-Renyi model for random graphs. Thus the findings allow to define a percolationbased index for almost ultrametricity which can be estimated in O(N2 logN) time which is more efficient than most ultrametricity indices.

We present a new proof of Harish-Chandra's formula $$\Pi(h_1) \Pi(h_2) \int_G e^{\langle \mathrm{Ad}_g h_1, h_2 \rangle} dg = \frac{ [ \! [ \Pi, \Pi ] \!] }{|W|} \sum_{w \in W} \epsilon(w) e^{\langle w(h_1),h_2 \rangle},$$ where $G$ is a compact, connected, semisimple Lie group, $dg$ is normalized Haar measure, $h_1$ and $h_2$ lie in a Cartan subalgebra of the complexified Lie algebra, $\Pi$ is the discriminant, $\langle \cdot, \cdot \rangle$ is the Killing form, $[ \! [ \cdot, \cdot ] \!]$ is an inner product that extends the Killing form to polynomials, $W$ is a Weyl group, and $\epsilon(w)$ is the sign of $w \in W$. The proof in this paper follows from a relationship between heat flow on a semisimple Lie algebra and heat flow on a Cartan subalgebra, extending methods developed by Itzykson and Zuber for the case of an integral over the unitary group $U(N)$. The heat-flow proof allows a systematic approach to studying the asymptotics of orbital integrals over a wide class of groups., Comment: 16 pages v4: Included edits that appeared in the published version, corrected minor typos, simplified proof of Lemma 2, small improvements to the text

This work is one of many that are devoted to the further investigation of local interpolating polynomial splines of the fifth order approximation. Here, new polynomial and trigonometrical basic splines are presented. The main features of these splines are the following: the approximation is constructed separately for each grid interval (or elementary rectangular), the approximation constructed as the sum of products of the basic splines and the values of function in nodes and/or the values of its derivatives and/or the values of integrals of this function over subintervals. Basic splines are determined by using a solving system of equations which are provided by the set of functions. It is known that when integrals of the function over the intervals is equal to the integrals of the approximation of the function over the intervals then the approximation has some physical parallel. The splines which are constructed here satisfy the property of the fifth order approximation. Here, the one-dimensional polynomial and trigonometrical basic splines of the fifth order approximation are constructed when the values of the function are known in each point of interpolation. For the construction of the spline, we use the discrete analogues of the first derivative and quadrature with the appropriate order of approximation. We compare the properties of these splines with splines which are constructed when the values of the first derivative of the function are known in each point of interpolation and the values of integral over each grid interval are given. The one-dimensional case can be extended to multiple dimensions through the use of tensor product spline constructs. Numerical examples are represented.

A ring extension $$R\subseteq S$$ is said to be residually FCP if, for all $$Q\in \mathrm {Spec}(S)$$, each chain of rings between $$R/(Q\cap R)$$ and S / Q is finite. The aim of this note is to establish several necessary and sufficient conditions for such extensions. We also study them, with emphasis on base rings R of Krull dimension 0. In particular we show that if $$R\subseteq S$$ is residually FCP and $$\dim (R)=0$$, then S is integral over R. This generalizes a result due to Anderson and Dobbs (Math. Rep. (Bucur.) 3(53):95–103, 2001).

The term of generalized material is introduced here as the material, whose state is described by the second order homogeneous differential equations with constant coefficients. The generalized point sources are described by homogeneous expressions, containing the first order derivatives with constant coefficients. The Green’s function for a half-space, made of generalized material, subjected to the action of generalized point sources, is derived in the form of a single integral over a circle. Some of the components of the surface Green’s function are presented in finite form, no computation of any integral is needed. The bonded contact problem is described as mixed–mixed boundary value problem for a half-space, with normal and tangential displacements prescribed inside domain S and vanishing tractions outside this domain on the plane $$x_{3}=0$$ . A set of governing integral equations for bonded contact problem was derived, with some of the kernels defined in finite form. The general crack problem is defined as that of a flat crack of arbitrary shape S in the plane $$x_{3}=0$$ , with normal tractions $$\sigma _{33}$$ applied symmetrically to the crack faces and tangential tractions $$\sigma _{31}$$ and $$\sigma _{23}$$ applied anti-symmetrically to the crack faces. A set of 3 governing integral equations is derived, with some of the kernels presented in the finite form. The relationship between the integrands in Fourier transform of the kernels of the sets of the integral equations of both problems is established: they are related in such a way, as if they were the coefficients in the schematic sets of algebraic equations. This relationship can also be presented as a general relationship between matrices of arbitrary rank n, with their components being determinants of certain matrices of rank q, with $$q\ge n$$ .

Purpose. The relevance of the scientific appeal to the problems of the quality of education in general and higher education in particular is essential for all subjects of the educational process. The final result and the achievement of the goals of education directly depend on the expansion of the criteria and approaches to an objective assessment of the quality of the teacher’s activity, and, in consequence, to improving his educational and methodological component of his work. The novelty of the study lies in the analysis of the quality of the educational process taking into account the views of students as direct participants in this process. Methods: the methodological basis of research is the dialectical method of cognition of reality, as well as a set of general scientific and private scientific methods of cognition; especially widely is used the methodology of specific social legal studies (observation, questionnaires, analysis of written sources, and methods of summarizing information, etc.). Results. The results of the research showed that the questioning of students is a necessary structural element of objective assessment system of education’s quality in the university, which should be used with a certain frequency, improving the survey methodology and respecting the moral and ethical standards in selecting indicators and scoring. Monitoring the quality of higher education through questioning students requires a scientific approach and generally affects the effectiveness of education. Scientific and practical significance. Conclusions can be used to create and upgrade local normative acts of higher education institutions, as well as conducting comprehensive studies of the quality of higher legal education and ways of its increase.

We study two relevant characterizations of a commutative positive operator valued measure (POVM) F. The first one is a Choquet type of an integral representation. It introduces a measure ν on the space of the projection valued measures (PVMs) and describes F as an integral over this space. The second one represents a commutative POVM F as the randomization of a single PVM E by means of a Markov kernel μ. We show that one can be derived from the other. We also elaborate upon some previous results on Choquet’s representation of Markov kernels and find a functional relationship between ν and μ. Finally, we analyze some relevant particular cases and provide some physically relevant examples which include the unsharp position observables.