Non-elementary Function Research Articles

ASSESSMENT OF MATERIAL DEFORMATION RESISTANCE IN BLANKS DURING DIRECT EXTRUSION USING THE DIE FORMING METHOD

In this scientific article, two deformation models for analyzing the accumulation of material damage during upsetting and direct extrusion using the die forming method are investigated. Both models include an analytical representation of the deformation trajectory, approximation of the boundary deformation curve, and a damage accumulation model. A three-parameter representation of the boundary deformation curve is applied, utilizing a power-law model and a linear damage accumulation model. The process of direct extrusion is examined by constructing parametric equations for the deformation trajectory. The functional dependence for the stress state indicator involves three constants, while a non-elementary function with a single constant is used for the accumulated deformation, determined using the theory of limit functions. Considering the complexity of trajectories, the use of a power-law accumulation model with a fixed nonlinearity indicator value is justified. The methodology for determining this indicator confirms its positive values less than one for the investigated trajectories, affirming its applicability. Numerical experiments demonstrate an increasing dependence of the limiting deformation with the increase of this indicator, indicating the effectiveness of the model.

Lower bounds on the state complexity of population protocols

Population protocols are a model of computation in which an arbitrary number of indistinguishable finite-state agents interact in pairs. The goal of the agents is to decide by stable consensus whether their initial global configuration satisfies a given property, specified as a predicate on the set of configurations. The state complexity of a predicate is the number of states of a smallest protocol that computes it. Previous work by Blondin et al. has shown that the counting predicates x ge eta have state complexity mathcal {O}(log eta ) for leaderless protocols and mathcal {O}(log log eta ) for protocols with leaders. We obtain the first non-trivial lower bounds: the state complexity of x ge eta is Omega (log log eta ) for leaderless protocols, and the inverse of a non-elementary function for protocols with leaders.

Entanglement harvesting of inertially moving Unruh-DeWitt detectors in Minkowski spacetime

We investigate the effects of relative motion on entanglement harvesting by considering a pair of Unruh-Dewitt detectors moving at arbitrary but independent and constant velocities, both linearly interacting with a massless scalar field vacuum in four-dimensional Minkowski spacetime. Working within the weak coupling approximation, we find that the negativity is a (nonelementary) function of the relative velocity of the detectors, as well as their energy gaps and minimal separation. We find parameter regions where negativity increases with velocity up to a maximum and then decreases, reaching zero at some sublight velocity. At any given relative velocity, the harvested entanglement is inversely correlated with the detector energy gap (at sufficiently high values) and the distance of closest approach of two detectors.

An Attempt to Make Non-Elementary Functions That Are Giving Solutions to Some Second-Order Nonlinear Autonomous ODEs

In this paper, we define an exponential function whose exponent is the product of a real number and the upper limit of integration in a non-elementary integral that can be arbitrary. We are using Abel’s methods, described by Armitage and Eberlein. The key is to start with a non-elementary integral function, differentiating and inverting, and then define a set of functions. Differentiating these functions twice give second-order nonlinear ODEs that have the defined set of functions as solutions.

Twin-width I: Tractable FO Model Checking

Inspired by a width invariant defined on permutations by Guillemot and Marx [SODA’14], we introduce the notion of twin-width on graphs and on matrices. Proper minor-closed classes, bounded rank-width graphs, map graphs, K t -free unit d -dimensional ball graphs, posets with antichains of bounded size, and proper subclasses of dimension-2 posets all have bounded twin-width. On all these classes (except map graphs without geometric embedding) we show how to compute in polynomial time a sequence of d -contractions , witness that the twin-width is at most d . We show that FO model checking, that is deciding if a given first-order formula ϕ evaluates to true for a given binary structure G on a domain D , is FPT in |ϕ| on classes of bounded twin-width, provided the witness is given. More precisely, being given a d -contraction sequence for G , our algorithm runs in time f ( d ,|ϕ |) · |D| where f is a computable but non-elementary function. We also prove that bounded twin-width is preserved under FO interpretations and transductions (allowing operations such as squaring or complementing a graph). This unifies and significantly extends the knowledge on fixed-parameter tractability of FO model checking on non-monotone classes, such as the FPT algorithm on bounded-width posets by Gajarský et al. [FOCS’15].

Guessing fractions of online sequences

An online algorithm is typically unaware of the length of the input request sequence that it is called upon. Consequently, it cannot determine whether it has already processed most of its input or whether the bulk of work is still ahead.In this paper, we are interested in whether some sort of orientation within the request sequence is nevertheless possible. For a real number 0<f<1, our objective is to preemptively guess the position fn within the request sequence of unknown length n: While processing the input, the online algorithm maintains a guess and is only allowed to update its guess to the position of the current element under investigation.We give a complete characterization of the optimal guessing strategies with respect to f: For f≤f0≈0.1867, the trivial algorithm that never updates its initial guess is (asymptotically) optimal. For f>f0, we give a randomized algorithm that in expectation places the guess at distance β(f)n from fn, where β is a non-elementary function, and we prove that this is optimal. Our findings show that the position fmaxn≈0.2847n is hardest to guess. We also give upper and lower bounds for a natural extension to weighted sequences.Our work can also be seen as the first randomized approach to the online checkpointing problem.

Constructing and Extending Description Logic Ontologies using Methods of Formal Concept Analysis

My thesis describes how methods from Formal Concept Analysis can be used for constructing and extending description logic ontologies. In particular, it is shown how concept inclusions can be axiomatized from data in the description logics mathcal {E}mathcal {L}, mathcal {M}, textsf {Horn}-mathcal {M}, and textsf{Prob}text{-}mathcal {E}mathcal {L}. All proposed methods are not only sound but also complete, i.e., the result not only consists of valid concept inclusions but also entails each valid concept inclusion. Moreover, a lattice-theoretic view on the description logic mathcal {E}mathcal {L} is provided. For instance, it is shown how upper and lower neighbors of mathcal {E}mathcal {L} concept descriptions can be computed and further it is proven that the set of mathcal {E}mathcal {L} concept descriptions forms a graded lattice with a non-elementary rank function.

A memory efficient implementation scheme of Gauss error function in a Laguerre-Volterra network for neuroprosthetic devices.

Cognitive neural prosthesis is a manmade device which can be used to restore or compensate for lost human cognitive modalities. The generalized Laguerre-Volterra (GLV) network serves as a robust mathematical underpinning for the development of such prosthetic instrument. In this paper, a hardware implementation scheme of Gauss error function for the GLV network targeting reconfigurable platforms is reported. Numerical approximations are formulated which transform the computation of nonelementary function into combinational operations of elementary functions, and memory-intensive look-up table (LUT) based approaches can therefore be circumvented. The computational precision can be made adjustable with the utilization of an error compensation scheme, which is proposed based on the experimental observation of the mathematical characteristics of the error trajectory. The precision can be further customizable by exploiting the run-time characteristics of the reconfigurable system. Compared to the polynomial expansion based implementation scheme, the utilization of slice LUTs, occupied slices, and DSP48E1s on a Xilinx XC6VLX240T field-programmable gate array has decreased by 94.2%, 94.1%, and 90.0%, respectively. While compared to the look-up table based scheme, 1.0×1017 bits of storage can be spared under the maximum allowable error of 1.0×10-3. The proposed implementation scheme can be employed in the study of large-scale neural ensemble activity and in the design and development of neural prosthetic device.

An improved homomorphism preservation theorem from lower bounds in circuit complexity

Previous work of the author [Rossman 2008a] showed that the Homomorphism Preservation Theorem of classical model theory remains valid when its statement is restricted to finite structures. In this paper, we give a new proof of this result via a reduction to lower bounds in circuit complexity, specifically on the AC 0 formula size of the colored subgraph isomorphism problem. Formally, we show the following: if a first-order sentence Φ of quantifier-rank k is preserved under homomorphisms on finite structures, then it is equivalent on finite structures to an existential-positive sentence Ψ of quantifier-rank k O (1) . Quantitatively, this improves the result of [Rossman 2008a], where the upper bound on the quantifier-rank of Ψ is a non-elementary function of k.

Возрастающие факториальные степени и неэлементарная функция типа интеграла Доусона

Возрастающие факториальные степени и неэлементарная функция типа интеграла Доусона

On the Evaluation of the Complex-Valued Exponential Integral

Abstract Although its applications span a broad scope of scientific fields ranging from applied physics to computer graphics, the exponential integral is a nonelementary special function available in specialized software packages but not in standard libraries, consequently requiring custom implementations on most platforms. In this paper, we provide a concise and comprehensive description of how to evaluate the complex-valued exponential integral. We first introduce some theoretical background on the main characteristics of the function, and outline available third-party proprietary implementations. We then provide an analysis of the various known representations of the function and present an effective algorithm allowing the computation of results within a desired accuracy, together with the corresponding pseudocode in order to facilitate portability onto various systems. An application to the calculation of the closed-form solution to single light scattering in homogeneous participating media illustrate...

The Word Problem in the Baumslag group with a non-elementary Dehn function is polynomial time decidable

We prove that the Word Problem in the Baumslag group G ( 1 , 2 ) = 〈 a , b ; a a b = a 2 〉 which has a non-elementary Dehn function is decidable in polynomial time.

On a nonelementary progress curve equation and its application in enzyme kinetics.

The analytical equation describing progress curves of an enzyme catalyzed reaction acting upon the Michaelis-Menten mechanism has been known for the case in which only the free enzyme incurs a loss of its activity, either spontaneously or as a result of an irreversible inhibitor action. The solution of differential equations which defines the rates of enzyme inactivation and substrate utilization is expressed by a nonelementary function in equation of an implicit type that precludes direct calculation of the extent of reaction at any time. Previously, the implicit equations have been rearranged to the alternative formulas and solved by the Newton-Raphson method, but this procedure may fail when used upon the presented equation. For this reason the other root-finding numerical method was applied, and the enzyme kinetic parameters of such numerically solved implicit equation for the reaction mechanism of irreversibly inhibited acetylcholinesterase were fitted to the experimental data by a nonlinear regression computer program.

A note on the homology covering of analytically finite klein surfaces

In previous works we have seen that a finitely generated torsion-free non-elementary function group is uniquely determined by its commutator subgroup. In this note, we observe that under the presence of orientation-reversing conformal automorphisms the above rigidity property still valid. More precisely, we see that finitely generated torsion-free reversing Fuchsian groups of the first kind are uniquely determined by their commutator subgroup. The arguments of the proof follow the same lines as for the orientable situation.

A note on the homology covering of closed Klein surfaces

In previous works we have seen that a finitely generated torsion-free non-elementary function group is uniquely determined by its commutator subgroup. In this note, we observe that under the presence of orientation-reversing conformal automorphisms the above rigidity property still valid. More precisely, we see that finitely generated torsion-free reversing Fuchsian groups of the first kind, without parabolic transformations, are uniquely determined by their commutator subgroup. The arguments of the proof follows the same lines as for the orientable situation.

Reduction Numbers of Ideals

We give general bounds for the reduction numbers of ideals in arbitrary Noetherian rings and multiplicity-dependent bounds for m-primary ideals in a Noetherian local ring (R,m). In the case of polynomial rings over fields the bound is a non-elementary function with four levels of exponentiation; for primary ideals the bound is linear in the Samuel multiplicity of the ideal. Finally we extend these techniques to generic complete intersections of dimension one.

Single-vacancy induced motion of a tracer particle in a two-dimensional lattice gas

We study the random motion of a tracer particle in a two-dimensional dense lattice gas. Repeated encounters of asingle vacancy displace the tracer particle from its initial position by a vector y of which we calculate the time-dependent distributionPt(y). On an infinite lattice and for large times $$P_t (y) \simeq \frac{{2(\pi - 1)}}{{\ln t}}K_0 \left( {\left( {\frac{{4\pi (\pi - 1)}}{{\ln t}}} \right)^{1/2} y} \right)$$ whereK0 is a modified Bessel function. The same problem is studied on a finiteL×L lattice with periodic boundary conditions; therePt(y) is shown to be a Gaussian on a time scaleL2 InL. On an ∞×L strip and for large times,Pt(y) is an explicitly given (but nonelementary) function of the scaling variable ξy1/t1/4, identical to the function occurring in the problem of a random walker on a random one-dimensional path.

Detection of cystic fibrosis heterozygotes using a modified loading with bromide.

A three day loading with sodium bromide was performed in 19 healthy controls, 16 definite cystic fibrosis heterozygotes, and 14 homozygote patients with cystic fibrosis. After three days sodium, potassium, chloride, and bromide were determined in serum and sweat. Using a multivariate discriminant analysis two nonelementary functions with the features sodium index, bromide index, potassium index, and chloride in sweat allowed us to identify the three groups. The reclassification of the probands, however, was correct in only 69.3%. A second analysis with the groups of controls and heterozygotes only resulted in a unidimensional nonelementary function with the features potassium index, bromide/sodium in sweat, and sodium index. The reclassification of the probands was correct in 91.4%. The discriminant values were 8.28 +/- 0.96 in the controls and 11.67 +/- 1.03 in the heterozygotes with an overlap of twice the standard deviations between 9.61 and 10.20.

Analytic properties of the vertex function in gauge theories. II

The analytic properties of the three-gluon vertex function for quantum chromodynamics in covariant gauges are investigated. First, a general tensor form for the vertex consistent with the Ward identity and free of kinematic singularities is constructed. The vertex is then calculated to one-loop order in the Feynman gauge. The complete expression for the off-shell one-loop vertex is expressed in terms of elementary functions plus one nonelementary function, the dilogarithm. Various kinematic limits of the vertex are considered. The most interesting results are the following. (1) Gluon mass-shell singularities occur in the transverse terms as well as the longitudinal terms. (2) The leading IR singularity is in the longitudinal part of the vertex, as is the case for QED; however, it is a pole singularity rather than the usual logarithmic singularity.

Analytic properties of the vertex function in gauge theories. I

The analytic properties of the vertex functions for massless scalar and massless and massive spinor electrodynamics are investigated. First, a general tensor form for the vertex consistent with the Ward identities and free of kinematic singularities is constructed. All of the scalar functions that appear are then calculated to one-loop order in perturbation theory. It is found that in both massive and massless theories the infrared singularities appear in the longitudinal parts of the vertex. The massless theories have additional photon mass-shell singularities which appear in the transverse terms. Complete expressions for each vertex are obtained which contain only one nonelementary function.