Nonconstant Function Research Articles

On the logical substantiveness of compositionality

AbstractGiven any set E of expressions freely generated from a set of atoms by syntactic operations, there exist trivially compositional functions on E (to wit, the injective and the constant functions), but also plenty of non-trivially compositional functions. Here we show that within the space of non-injective functions (and so a fortiori within the space of non-injective and non-constant functions), compositional functions are not sufficiently abundant in order to generate the consequence relation of every propositional logic. Logical consequence relations thus impose substantive constraints on the existence of compositional functions when coupled with the condition of non-injectivity (though not without it). We ask how the apriori exclusion of injective functions from the search space might be justified, and we discuss the prospects of claims to the effect that any function can be “encoded” in a compositional one.

Every exponential group supports a positive harmonic function

We prove that all groups of exponential growth support non-constant positive harmonic functions. In fact, our results hold in the more general case of strongly connected, finitely supported Markov chains invariant under some transitive group of automorphisms for which the directed balls grow exponentially.

Testing for Equal Average Forecast Accuracy in Possibly Unstable Environments *

We consider the issue of testing the null of equal average forecast accuracy in a model where the forecast error loss differential series has a potentially non-constant mean function over time. We show that when time variation is present in the loss differential mean, the standard Diebold and Mariano (1995) test, which was proposed for evaluating forecasts in a stable environment, has an asymptotic size of zero, and, whilst consistent, can have reduced local power. This arises due to inconsistent estimation of the implicit long run variance estimator, which diverges under a time varying mean. We suggest a modified statistic that replaces the standard long run variance estimator based on full-sample demeaning of the loss differential series with one based on nonparametric local demeaning. The new long run variance estimator is consistent under both the null and alternative when the mean function is time varying or constant, and in both cases, the modified test recovers the asymptotic size and power properties associated with the original test in the constant mean case. The modified test therefore provides a robust method for testing the equal average forecast accuracy null, allowing for instability in the loss differential mean. The benefits of our test are demonstrated via Monte Carlo simulation and two empirical applications.

On manifolds with nonnegative Ricci curvature and the infimum of volume growth order < 2

Abstract We study open (complete and noncompact) n-manifolds M with nonnegative Ricci curvature, under the condition that any asymptotic cone of M splits off an ℝ k {\mathbb{R}^{k}} factor. In particular, we obtain two rigidity results for open n-manifolds M with Ric M ≥ 0 {\mathrm{Ric}_{M}\geq 0} and the infimum of volume growth order < 2 {<2} . The first result asserts that there exists a nonconstant linear growth harmonic function on M if and only if M is isometric to the metric product ℝ × N {\mathbb{R}\times N} for some compact manifold N. The second asserts that the Riemannian universal cover of M has Euclidean volume growth if and only if M is flat with an n - 1 {n-1} dimensional soul.

Developing novel wormhole metrics in finsler-randers geometry using the barthel connection and osculating-riemannian method

In this research, we focus on modelling a wormhole using Finsler-Randers geometry. By applying the Barthel connection from the range of Finslerian connections, we have developed a new wormhole metric, termed the osculating Barthel-Finsler-Randers metric, through the osculating-Riemannian method within the Finsler-Randers framework. The fundamental field equations are formulated in terms of the obtained metric tensor characterised by η(r), which plays a pivotal role in characterizing the anisotropy of the model. To explore the application of Finsler geometry in addressing the existence of a wormhole, we adopt an exponential shape function. Further, in order to streamline the computational aspects of our analysis, and to discuss the effect of anisotropy effect of Finsler Rander geometry, we adopt a specific functional form for the anisotropic parameter η(r)=arn, where α is a constant with implications for the anisotropy, among various possibilities for η(r) a non-zero analytical component of vector field in Randers metric. In addition, we consider three distinct values for the exponent n = −1, 0, and 1 and subsequently, we proceed to develop two distinct models for each of these three values of n with constant and non-constant redshift function. We analyse the physical implications of η(r) on the energy conditions. Our results demonstrate the crucial impact of anisotropy on the stability and structure of wormholes, emphasizing the requirement of exotic matter under specific conditions. This research deepens the understanding of wormhole physics within Finsler geometry, shedding light on the theoretical frameworks essential for the existence and stability of these cosmic structures.

Topological rigidity of maps in positive characteristic and anabelian geometry

We study pairs of non-constant maps between two integral schemes of finite type over two (possibly different) fields of positive characteristic. When the target is quasi-affine, Tamagawa showed that the two maps are equal up to a power of Frobenius if and only if they induce the same homomorphism on their étale fundamental groups. We extend Tamagawa’s result by adding a purely topological criterion for maps to agree up to a power of Frobenius.

Non-constant functions with zero nonlocal gradient and their role in nonlocal Neumann-type problems

This work revolves around properties and applications of functions whose nonlocal gradient, or more precisely, finite-horizon fractional gradient, vanishes. Surprisingly, in contrast to the classical local theory, we show that this class forms an infinite-dimensional vector space. Our main result characterizes the functions with zero nonlocal gradient in terms of two simple features, namely, their values in a layer around the boundary and their average. The proof exploits recent progress in the solution theory of boundary-value problems with pseudo-differential operators. We complement these findings with a discussion of the regularity properties of such functions and give illustrative examples. Regarding applications, we provide several useful technical tools for working with nonlocal Sobolev spaces when the common complementary-value conditions are dropped. Among these, are new nonlocal Poincaré inequalities and compactness statements, which are obtained after factoring out functions with vanishing nonlocal gradient. Following a variational approach, we exploit the previous findings to study a class of nonlocal partial differential equations subject to natural boundary conditions, in particular, nonlocal Neumann-type problems. Our analysis includes a proof of well-posedness and a rigorous link with their classical local counterparts via Γ-convergence as the fractional parameter tends to 1.

On Splitting Complete Manifolds via Infinity Harmonic Functions

Abstract In this paper, we prove some splitting results for manifolds supporting a non-constant infinity harmonic function which has at most linear growth on one side. Manifolds with non-negative Ricci or sectional curvature are considered. In dimension $2$, we extend Savin’s theorem on Lipschitz infinity harmonic functions in the plane to every surface with non-negative sectional curvature.

Cosmological constraints on f(Q) gravity models in the non-coincident formalism

The article investigates cosmological applications of f(Q) theories in a non-coincident formalism. We explore a new f(Q) theory dynamics utilizing a non-vanishing affine connection involving a non-constant function γ(t)=−a−1H˙, resulting in Friedmann equations that are entirely distinct from those of f(T) theory. In addition, we propose a new parameterization of the Hubble function that can consistently depicts the present deceleration parameter value, transition redshift, and the late time de-Sitter limit. We evaluate the predictions of the assumed Hubble function by imposing constraints on the free parameters utilizing Bayesian statistical analysis to estimate the posterior probability by employing the CC, Pantheon+SH0ES, and the BAO samples. Moreover, we conduct the AIC and BIC statistical evaluations to determine the reliability of MCMC analysis. Further, we consider some well-known corrections to the STEGR case such as an exponentital f(Q) correction, logarithmic f(Q) correction, and a power-law f(Q) correction and then we find the constraints on the parameters of these models via energy conditions. Finally, to test the physical plausibility of the assumed f(Q) models we conduct the thermodynamical stability analysis via the sound speed parameter.

Geometrical Analysis of Spacelike and Timelike Rectifying Curves and Their Applications

In the light of great importance of curves and their frames in many different branches of science, especially differential geometry as well as geometric properties and their uses in various fields, we are interested here to study a special kind of curves called rectifying curves. We consider some characterizations of a non-lightlike curve has a spacelike or timelike rectifying plane in pseudo-Euclidean space E13. Then, we demonstrate that the proportion of curvatures of any spacelike or timelike rectifying curve is a non-constant linear function of the arc length parameter s. Finally, we defray a computational example to support our main findings.

Recurrent and (strongly) resolvable graphs

We develop a new approach to recurrence and the existence of non-constant harmonic functions on infinite weighted graphs. The approach is based on the capacity of subsets of metric boundaries with respect to intrinsic metrics. The main tool is a connection between polar sets in such boundaries and null sets of paths. This connection relies on suitably diverging functions of finite energy.

Traversable wormholes from Loop Quantum Gravity

This study introduces and investigates Lorentzian traversable wormhole solutions rooted in Loop Quantum Gravity (LQG). The static and spherically symmetric solutions to be examined stem from the energy density sourcing self-dual regular black holes discovered by L. Modesto, relying on the parameters associated with LQG, which account for the quantum nature of spacetime. We specifically focus on macroscopic wormholes characterized by small values of these parameters. Our analysis encompasses zero-tidal solutions and those with non-constant redshift functions, exploring immersion diagrams, curvatures, energy conditions, equilibrium requirements, and the requisite quantity of exotic matter to sustain these wormholes. The investigation underscores the influence of LQG parameters on these features, highlighting the pivotal role of spacetime's quantum properties in shaping such objects and governing their behavior.

Line bundles on the moduli space of Lie algebroid connections over a curve

We explore algebro-geometric properties of the moduli space of holomorphic Lie algebroid (L) connections on a compact Riemann surface X of genus g≥3. A smooth compactification of the moduli space of L-connections, such that underlying vector bundle is stable, is constructed; the complement of the moduli space inside the compactification is a divisor. A criterion for the numerical effectiveness of the boundary divisor is given. We compute the Picard group of the moduli space, and analyze Lie algebroid Atiyah bundles associated with an ample line bundle. This enables us to conclude that regular functions on the space of certain Lie algebroid connections are constants. Moreover, under some condition, it is shown that the moduli space of L-connections does not admit non-constant algebraic functions. Rationally connectedness of the moduli spaces is explored.

Some Properties of the Functions Representable as Fractional Power Series

The α-fractional power moduli series are introduced as a generalization of α-fractional power series and the structural properties of these series are investigated. Using the fractional Taylor’s formula, sufficient conditions for a function to be represented as an α-fractional power moduli series are established. Beyond theoretical formulations, a practical method to represent solutions to boundary value problems for fractional differential equations as α-fractional power series is discussed. Finally, α-analytic functions on an open interval I are defined, and it is shown that a non-constant function is α-analytic on I if and only if 1/α is a positive integer and the function is real analytic on I.

Interior Multi-Peak Solutions for a Slightly Subcritical Nonlinear Neumann Equation

In this paper, we consider the nonlinear Neumann problem (Pε): −Δu+V(x)u=K(x)u(n+2)/(n−2)−ε, u>0 in Ω, ∂u/∂ν=0 on ∂Ω, where Ω is a smooth bounded domain in Rn, n≥4, ε is a small positive real, and V and K are non-constant smooth positive functions on Ω¯. First, we study the asymptotic behavior of solutions for (Pε) which blow up at interior points as ε moves towards zero. In particular, we give the precise location of blow-up points and blow-up rates. This description of the interior blow-up picture of solutions shows that, in contrast to a case where K≡1, problem (Pε) has no interior bubbling solutions with clustered bubbles. Second, we construct simple interior multi-peak solutions for (Pε) which allow us to provide multiplicity results for (Pε). The strategy of our proofs consists of testing the equation with vector fields which make it possible to obtain balancing conditions which are satisfied by the concentration parameters. Thanks to a careful analysis of these balancing conditions, we were able to obtain our results. Our results are proved without any assumptions of the symmetry or periodicity of the function K. Furthermore, no assumption of the symmetry of the domain is needed.

Ricci Vector Fields

We introduce a special vector field ω on a Riemannian manifold (Nm,g), such that the Lie derivative of the metric g with respect to ω is equal to ρRic, where Ric is the Ricci tensor of (Nm,g) and ρ is a smooth function on Nm. We call this vector field a ρ-Ricci vector field. We use the ρ-Ricci vector field on a Riemannian manifold (Nm,g) and find two characterizations of the m-sphere Smα. In the first result, we show that an m-dimensional compact and connected Riemannian manifold (Nm,g) with nonzero scalar curvature admits a ρ-Ricci vector field ω such that ρ is a nonconstant function and the integral of Ricω,ω has a suitable lower bound that is necessary and sufficient for (Nm,g) to be isometric to m-sphere Smα. In the second result, we show that an m-dimensional complete and simply connected Riemannian manifold (Nm,g) of positive scalar curvature admits a ρ-Ricci vector field ω such that ρ is a nontrivial solution of the Fischer–Marsden equation and the squared length of the covariant derivative of ω has an appropriate upper bound, if and only if (Nm,g) is isometric to m-sphere Smα.

Exceptional values of entire functions of finite order in one of the variables

Let F(z,w) be a holomorphic function in Cn×C of finite order in w with n≥2. Let Ω be the set of points z∈Cn where F(z,w) is a non-constant function omitting a value π(z). Near a finite accumulation point z0 of Ω, we prove in the main result (Theorem 1) that Ω is a local analytic set and π(z) is holomorphic, and show the existence of a proper globally analytic set Δ of Cn such that either Ω⊂Δ or Ω=Cn∖Δ, being possible in the last case to also determine F(z,w) in terms of π(z). We apply this result to several problems. First, we extend a Theorem due to Nishino about exceptional values when near z0 dimension of Ω is n and assure the existence of a meromorphic function α(z) in Cn such that π(z)=α(z) except at points where α(z) has poles or F(z,w) is constant (also being F(z,w) a polynomial in w if α(z) is ∞). After, we prove that Ω is a local analytic set in Cn and the existence of a proper analytic subset E of Cn such that Ω⊂E or Ω=Cn∖E. Finally, we generalize a Lelong-Gruman Theorem about the set of points z where π(z)=0.

A new aspect of Chebyshev’s bias for elliptic curves over function fields

This work addresses the prime number races for non-constant elliptic curves E E over function fields. We prove that if r a n k ( E ) > 0 \mathrm {rank}(E) > 0 , then there exist Chebyshev biases towards being negative, and otherwise there exist Chebyshev biases towards being positive. The key input is the convergence of the partial Euler product at the centre, which follows from the Deep Riemann Hypothesis over function fields.

Functions on the moduli space of projective structures on complex curves

We investigate the moduli space Pg of smooth complex projective curves of genus g equipped with a projective structure. When g≥3, it is shown that this moduli space Pg does not admit any nonconstant algebraic function. This is in contrast with the case of P2 which is known to be an affine variety.

On special values of generators of modular function fields

Let N be a positive integer and ζN be a primitive Nth root of unity. Let X(N) be the modular curve of level N and FX(N),Q(ζN) be its field of meromorphic functions with Fourier coefficients in Q(ζN). Let u and v be nonconstant functions in FX(N),Q(ζN) which generate a sufficiently large subfield of FX(N),Q(ζN) over Q(ζN). In this paper, we give sufficient conditions for generating the ray class field of an imaginary quadratic field K modulo N over K(ζN) by the special values of u and v.