Finite Part Research Articles

“POLYPREDICATIVE CONSTRUCTION” IN THE KUMYK LANGUAGE

The article examines the functioning of the polypredicative construction in the Kumyk language. The notion of a "polypredicative sentence" was introduced by the Novosibirsk syntactic school. Turkic languages are not characterized by properly complex sentences with two formally independent finite parts connected by an analytical form. Case affixes and postpositions form not finite verb forms, but infinite verb forms. Such constructions in agglutinative languages are the most frequent.

Feedback stabilization of parabolic systems with input delay

<p style='text-indent:20px;'>This work is devoted to the stabilization of parabolic systems with a finite-dimensional control subjected to a constant delay. Our main result shows that the Fattorini-Hautus criterion yields the existence of such a feedback control, as in the case of stabilization without delay. The proof consists in splitting the system into a finite dimensional unstable part and a stable infinite-dimensional part and to apply the Artstein transformation on the finite-dimensional system to remove the delay in the control. Using our abstract result, we can prove new results for the stabilization of parabolic systems with constant delay: the <inline-formula><tex-math id="M1">\begin{document}$ N $\end{document}</tex-math></inline-formula>-dimensional linear reaction-convection-diffusion equation with <inline-formula><tex-math id="M2">\begin{document}$ N\geq 1 $\end{document}</tex-math></inline-formula> and the Oseen system. We end the article by showing that this theory can be used to stabilize nonlinear parabolic systems with input delay by proving the local feedback distributed stabilization of the Navier-Stokes system around a stationary state.</p>

Приближенное решение гиперсингулярного интегрального уравнения I рода, ограниченное на обоих концах отрезка интегрирования, с применением рядов Чебышева

A hypersingular integral equation on the interval of integration is considered. The hypersingular integral is understood in the sense of Hadamard, that is, in the finite part. The class of such equations is widely used in problems of mathematical physics, in technology, and most importantly: in recent years, they are one of the main devices for modeling problems in electrodynamics. With the use of Chebyshev polynomials of the second kind, the unknown function, the right-hand side and the kernel are replaced in the equation. The expansion coefficients of these functions are calculated using quadrature formulas of the highest algebraic degree of accuracy, i.e., Gauss quadrature formulas. Thus, the equation is discretized. The result is an infinite system of linear algebraic equations for the expansion coefficients of the unknown function. The fact that the hypersingular integral equation in the case under consideration has a unique solution in the class of sufficiently smooth functions is taken into account. The constructed computational scheme is substantiated using the general theory of functional analysis. The calculation error is estimated under certain conditions relative to the right-hand side and the kernel of the equation. The described method for solving the hypersingular integral equation is illustrated by test examples that show the high efficiency of the method.

Effective action for delta potentials: Spacetime-dependent inhomogeneities and Casimir self-energy

We study the vacuum fluctuations of a quantum scalar field in the presence of a thin and inhomogeneous flat mirror, modeled with a delta potential. Using heat-kernel techniques, we evaluate the Euclidean effective action perturbatively in the inhomogeneities (nonperturbatively in the constant background). We show that the divergences can be absorbed into a local counterterm and that the remaining finite part is in general a nonlocal functional of the inhomogeneities, which we compute explicitly for massless fields in $D=4$ dimensions. For time-independent inhomogeneities, the effective action gives the Casimir self-energy for a partially transmitting mirror. For time-dependent inhomogeneities, the Wick-rotated effective action gives the probability of particle creation due to the dynamical Casimir effect.

Conformal field theories on deformed spheres, anomalies, and supersymmetry

We study the free energy of four-dimensional CFTs on deformed spheres. For generic nonsupersymmetric CFTs only the coefficient of the logarithmic divergence in the free energy is physical, which is an extremum for the round sphere. We then specialize to N=2N=2 SCFTs where one can preserve some supersymmetry on a compact manifold by turning on appropriate background fields. For deformations of the round sphere the cc anomaly receives corrections proportional to the supersymmetric completion of the (Weyl)^22 term, which we determine up to one constant by analyzing the scale dependence of various correlators in the stress-tensor multiplet. We further show that the double derivative of the free energy with respect to the marginal couplings is proportional to the two-point function of the bottom components of the marginal chiral multiplet placed at the two poles of the deformed sphere. We then use anomaly considerations and counter-terms to parametrize the finite part of the free energy which makes manifest its dependence on the Kähler potential. We demonstrate these results for a theory with a vector multiplet and a massless adjoint hypermultiplet using results from localization. Finally, by choosing a special value of the hypermultiplet mass where the free energy is independent of the deformation, we derive an infinite number of constraints between various integrated correlators in N=4N=4 super Yang-Mills with any gauge group and at all values of the coupling, extending previous results.

ZH production in gluon fusion: two-loop amplitudes with full top quark mass dependence

We present results for the two-loop helicity amplitudes entering the NLO QCD corrections to the production of a Higgs boson in association with a Z -boson in gluon fusion. The two-loop integrals, involving massive top quarks, are calculated numerically. Results for the interference of the finite part of the two-loop amplitudes with the Born amplitude are shown as a function of the two kinematic invariants on which the amplitudes depend.

Regularity criterion for 3D nematic liquid crystal flows in terms of finite frequency parts in dot{B}_{infty,infty }^{-1}

In this paper, we establish the regularity criterion for the weak solution of nematic liquid crystal flows in three dimensions when the L^{infty }(0,T;dot{B}_{infty,infty }^{-1})-norm of a suitable low frequency part of (u,nabla d) is bounded by a scaling invariant constant and the initial data (u_{0},nabla d_{0}). Our result refines the corresponding one in (Liu and Zhao in J. Math. Anal. Appl. 407:557-566, 2013) and that in (Ri in Nonlinear Anal. TMA 190:111619, 2020).

An Ideal Class to Construct Solutions for Skew Brownian Motion Equations

This paper contributes to the study of stochastic processes of the class \((\Sigma )\). First, we extend the notion of the above-mentioned class to càdlàg semi-martingales, whose finite variation part is considered càdlàg instead of continuous. Thus, we present some properties and propose a method to characterize such stochastic processes. Second, we investigate continuous processes of the class \((\Sigma )\). More precisely, we derive a series of new characterization results. In addition, we construct solutions for skew Brownian motion equations using continuous stochastic processes of the class \((\Sigma )\).

Renormalization of CP -violating nuclear forces

Electric dipole moments of nuclei, diamagnetic atoms, and certain molecules are induced by CP-violating nuclear forces. Naive dimensional analysis predicts these forces to be dominated by long-range one-pion-exchange processes, with short-range forces entering only at next-to-next-to-leading order in the chiral expansion. Based on renormalization arguments we argue that a consistent picture of CP-violating nuclear forces requires a leading-order short-distance operator contributing to ${}^1S_0$-${}^3P_0$ transitions, due to the attractive and singular nature of the strong tensor force in the ${}^3P_0$ channel. The short-distance operator leads to $\mathcal O(1)$ corrections to static and oscillating, relevant for axion searches, electric dipole moments. We discuss strategies how the finite part of the associated low-energy constant can be determined in the case of CP violation from the QCD theta term by the connection to charge-symmetry violation in nuclear systems.

Compatibility checking for asynchronously communicating software

• Inter-operability of distributed systems. • Asynchronous operational semantics using FIFO buffers. • Decidability of synchronizability and wellformedness. • Refinement based proofs and model checking of distributed systems' compatibility. • Tool support. Compatibility is a crucial problem that is encountered while constructing new software by reusing and composing existing components. A set of software components is called compatible if their composition preserves certain properties, such as deadlock freedom. However, checking compatibility for systems communicating asynchronously is an undecidable problem, and asynchronous communication is a common interaction mechanism used in building software systems. A typical approach in analyzing such systems is to bound the state space. In this article, we propose a different approach and do not impose any bounds on the number of participants or the sizes of the message buffers. Instead, we present a sufficient condition for checking compatibility of a set of asynchronously communicating components. Our approach relies on the synchronizability property which identifies systems for which interaction behavior remains the same when asynchronous communication is replaced with synchronous communication. Using the synchronizability property, we can check the compatibility of systems with unbounded message buffers by analyzing only a finite part of their behavior. We have implemented a prototype tool to automate our approach and we have applied it to many examples.

The Roadheader Auto-Rectification Dynamic Analysis and Control Based on the Roadway Floor Mechanic Characteristics

The auto-rectification of the roadheader is one of the most key steps in the roadway auto-cutting progress, and the performance of the roadheader rectification on the mechanized excavation faces the key performance parameter. The determination of the roadway floor mechanical parameter is the big challenge and difficulty to the automation excavation of the roadheader. Combined the component of the roadway floor mechanical property and experiment parameters for the mechanical parameters regression equation, the floor mechanical parameter formula can be got, which is accurate and could be utilized in calculating the resistance. Analyzing the finite element part on the compaction resistance and steering resistance of the roadheader track, the accurate and reliable dynamic model of the roadheader rectification can be built. Combined with the roadheader’s structure and motion characteristic, the neural net PID(Proportion Integration Differentiation)control method is proposed which can real-time adjust the control parameters. Then the dynamic model and control method are simulated by MATLAB and the control method is tested by the roadheader control experiment system. Based on the experiment result, the designed control system can well meet the control requirements, fast response, small overshoot and better stability, which could ensure the good dynamic performance of the roadheader and the cutting quality of the roadway. It could be the theoretical basis and important technical support for the intelligent mining of coal mine.

Divergent Integrals Treatment in Quantum Field Theory

This paper presents a brief review on regularization methods and shows that the combination of two techniques could preserve symmetries in all orders of the Perturbation Theory. We will see, with two simple examples in the one-loop-order, what a regularization method needs to preserve symmetries. Furthermore, it will be shown that the problems with symmetry violations by regularization methods are related to the consequences of not being able to simply shift a variable in these divergent integrals. To facilitate the analysis between methods and provide analytical expressions for the finite parts of amplitudes, it was systematized in terms of a set of special functions. In the Appendix we introduce these functions and then cases of specific interest, functions continuity and some useful limits will be presented.

Modern methods of calculations of bremsstrahlung in the interaction of elementary particles

. The problem of real bremsstrahlung calculation is considered using the modern methods of regularization of divergencies. In particular, we calculate soft photon bremsstrahlung in the most general form using the method of dimensional regularization of infrared divergences. The general calculation algorithm of hard photon bremsstrahlung is described. It is shown that the contribution of hard bremsstrahlung can be separated into the finite and divergent parts. The divergent part can be factorized with the contribution of the initial process in the Born approximation. It is shown that a good choice of kinematic variables makes an analytic covariant calculation of the divergent part of the hard bremsstrahlung possible. In a particular case, an algorithm for determining the kinematic constraints on the invariants is described. A numerical analysis of the radiative corrections for gauge bosons production processes in the case of electron-photon collisions is performed. It is discovered that the contribution of the finite part of bremsstrahlung at high collision energies reaches 20 per cent and must be taken into account in calculations of radiative corrections. The results obtained can be used in various calculations, including covariant ones, performed in the context of confirmation of the Standard Model theoretical predictions or searching for manifestations of alternative gauge models.

MONOFINITE COMPLEX CLAUSE IN THE KUMYK LANGUAGE

This article aims to analyze the monofinite complex clause in the Kumyk language in terms of structure. The originality of the research lies in the fact that the article proposes a different approach than in Russian studies on the analysis of the syntactic structure of a sentence. The relevance of the research is determined by the role of the sentence structure and methods of its study in linguistics in general. The main method used in the study is the method of syntactic modeling, which is used in terms of functions (members of a sentence), in terms of morphological forms of words and the semantic structure of a sentence. Turkic languages are not characterized by properly complex sentences with two formally independent finite parts connected by an analytical form. Case affixes and postpositions form not finite verb forms, but infinite verb forms. Such constructions in agglutinative languages are central and most frequent. The monofinite complex clause, or a sentence complicated by an infinite phrase (participial and adverbial phrase), or “polypredicative construction” according to the Novosibirsk syntactic school is investigated in this article. The author argues that when characterizing a complex clause, it is necessary to take into account such parameters as the finiteness / infiniteness of the dependent predicate, the nature of the means of communication and the referential identity / non-identity of the subjects of the main and dependent parts. The constructive center of a monofinite complex clause is the indicator of connection –analytical, that is, an auxiliary word, or synthetic, that is, a morpheme in the composition of a dependent predicate. Formally, this is a part of the infinite form of such a predicate, but functionally it serves to express the relationship between the predicative parts. Since the dependent predicate in the Kumyk language occupies the final position in the dependent predicative unit, the synthetic link indicator is located on the border between the main predicative unit and the dependent predicative unit.

ON A PROBLEM FOR AN ELASTIC INFINITE SHEET STRENGTHENED BY TWO PARALLEL STRINGERS WITH FINITE LENGTHS THROUGH ADHESIVE SHEAR LAYERS

The article considers the problem for an elastic infinite sheet (plate), which is strengthened on two parallel finite parts of its upper surface by two parallel finite stringers with different elastic properties. The parallel stringers are located asymmetrically with respect to the horizontal axis of the sheet and deform under the action of horizontal forces. The interaction between the infinite sheet and stringers takes place through thin elastic adhesive layers. The problem of determining unknown shear stresses acting between the infinite sheet and stringers is reduced to a system of Fredholm integral equations of second kind with respect to unknown functions, which are specified on two parallel finite intervals. It is shown that in the certain domain of the change of the characteristic parameters of the problem this system of integral equations in Banach space can be solved by the method of successive approximations. Particular cases are considered, the character and behaviour of unknown shear stresses are investigated.

Towards a power counting in nuclear energy–density–functional theories through a perturbative analysis

We illustrate a step towards the construction of a power counting in energy–density–functional (EDF) theories, by analyzing the equations of state (EOSs) of both symmetric and neutron matter. Within the adopted strategy, next–to–leading order (NLO) EOSs are introduced which contain renormalized first–order–type terms and an explicit second–order finite part. Employing as a guide the asymptotic behavior of the introduced renormalized parameters, we focus our analysis on two aspects: (i) With a minimum number of counterterms introduced at NLO, we show that each energy contribution entering in the EOS has a regular evolution with respect to the momentum cutoff (introduced in the adopted regularization procedure) and is found to converge to a cutoff–independent curve. The convergence features of each term are related to its Fermi–momentum dependence. (ii) We find that the asymptotic evolution of the second–order finite–part coefficients is a strong indication of a perturbative behavior, which in turns confirms that the adopted strategy is coherent with a possible underlying power counting in the chosen Skyrme–inspired EDF framework.

On the adjoint representation of a hopf algebra

AbstractWe consider the adjoint representation of a Hopf algebra $H$ focusing on the locally finite part, $H_{{\textrm ad\,fin}}$, defined as the sum of all finite-dimensional subrepresentations. For virtually cocommutative $H$ (i.e., $H$ is finitely generated as module over a cocommutative Hopf subalgebra), we show that $H_{{\textrm ad\,fin}}$ is a Hopf subalgebra of $H$. This is a consequence of the fact, proved here, that locally finite parts yield a tensor functor on the module category of any virtually pointed Hopf algebra. For general Hopf algebras, $H_{{\textrm ad\,fin}}$ is shown to be a left coideal subalgebra. We also prove a version of Dietzmann's Lemma from group theory for Hopf algebras.

The mass quadrupole moment of compact binary systems at the fourth post-Newtonian order

The mass-type quadrupole moment of inspiralling compact binaries (without spins) is computed at the fourth post-Newtonian (4PN) approximation of general relativity. The multipole moments are defined by matching between the field in the exterior zone of the matter system and the PN field in the near zone, following the multipolar-post-Minkowskian (MPM)-PN formalism. The matching implies a specific regularization for handling infra-red (IR) divergences of the multipole moments at infinity, based on the Hadamard finite part procedure. On the other hand, the calculation entails ultra-violet (UV) divergences due to the modelling of compact objects by delta-functions, that are treated with dimensional regularization (DR). In future work we intend to systematically study the IR divergences by means of dimensional regularization as well. Our result constitutes an important step in the goal of obtaining the gravitational wave templates of inspiralling compact binary systems with 4PN/4.5PN accuracy.

Analytic Continuation of Resolvents of Elliptic Operators in a Multidimensional Cylinder

We consider the scalar second order elliptic differential operator in a multidimensional cylinder with the Dirichlet boundary condition. The coefficients of the operator are periodic on both outlets of the cylinder and are arbitrary in its finite part. We study a local analytic continuation of the bordered resolvent of the operator from the upper half-plane to the lower one with respect to the spectral parameter in a neighborhood of an interior point of the essential spectrum. It is shown that the size of the neighborhood depends only on geometric properties of the cylinder and the behavior of periodic components of coefficients of the operator. We introduce the notion of a resonance and formulate the corresponding boundary value problems. We describe the behavior of the bordered resolvent with respect to the spectral parameter in the neighborhood of the point of the essential spectrum.

Logarithmic negativity in quantum Lifshitz theories

We investigate quantum entanglement in a non-relativistic critical system by calculating the logarithmic negativity of a class of mixed states in the quantum Lifshitz model in one and two spatial dimensions. In 1+1 dimensions we employ a correlator approach to obtain analytic results for both open and periodic biharmonic chains. In 2+1 dimensions we use a replica method and consider spherical and toroidal spatial manifolds. In all cases, the universal finite part of the logarithmic negativity vanishes for mixed states defined on two disjoint components. For mixed states defined on adjacent components, we find a non-trivial logarithmic negativity reminiscent of two-dimensional conformal field theories. As a byproduct of our calculations, we obtain exact results for the odd entanglement entropy in 2+1 dimensions.