Nontrivial Function Research Articles

Exploiting translational invariance in matrix product state simulations of spin chains with periodic boundary conditions

We present a matrix product state (MPS) algorithm to approximate ground states of translationally invariant systems with periodic boundary conditions. For a fixed value of the bond dimension $D$ of the MPS, we discuss how to minimize the computational cost to obtain a seemingly optimal MPS approximation to the ground state. In a chain with $N$ sites and correlation length $\ensuremath{\xi}$, the computational cost formally scales as $g(D,\ensuremath{\xi}/N){D}^{3}$, where $g(D,\ensuremath{\xi}/N)$ is a nontrivial function. For $\ensuremath{\xi}\ensuremath{\ll}N$, this scaling reduces to ${D}^{3}$, independent of the system size $N$, making our method $N$ times faster than previous proposals. We apply the algorithm to obtain MPS approximations for the ground states of the critical quantum Ising and Heisenberg spin-$1/2$ models as well as for the noncritical Heisenberg spin-$1$ model. In the critical case, for any chain length $N$, we find a model-dependent bond dimension $D(N)$ above which the polynomial decay of correlations is faithfully reproduced throughout the entire system.

New method and treatment technique applied to interband transition in GaAs1−x Px ternary alloys

Abstract In this paper we presented a new method (Eigen-Coordinates (ECs)) that can be used for calculations of the critical points (CPs) energy of the interband-transition edges of the heterostructures. This new method is more accurate and complete in comparison with conventional ones and has a wide range of application for the calculation of the fitting parameters related to nontrivial functions that initially have nonlinear fitting parameters that are difficult to evaluate. The new method was applied to determine the CPs energies from the dielectric functions of the MBE grown GaAs1−xPx ternary alloys obtained using spectroscopic ellipsometry (SE) measurements at room temperature in the 0.5-5 eV photon energy region. The obtained results are in good agreement with the results of the other methods.

Algorithms for special functions

In this thesis we present our contributions to symbolic summation, extending WZFasenmyer methods to handle definite hypergeometric sums with nonstandard boundary conditions and to compute recurrences for multiple Mellin-Barnes integrals over hypergeometric terms. We also include concrete applications of these methods to Feynman integral calculus, as well as for proving identities involving definite integrals and special functions. First we give a short introduction to WZ-summation methods, including K. Wegschaider’s approach to this method and its implementation in the package MultiSum. Inspired by work of Sister Celine Fasenmyer, these techniques were introduced by H. Wilf and D. Zeilberger to algorithmically compute recurrences for multiple sums over hypergeometric terms. Their procedure is based on finding a certificate recurrence satisfied by the hypergeometric summand and summing over this difference equation to obtain a recurrence for the nested sum. Our proofs of two nontrivial special function identities involving Gegenbauer polynomials, provide classic applications of the method. As part of the collaboration between RISC and DESY coordinated by C. Schneider, we developed an algorithmic approach to compute Feynman parameter integrals after rewriting them as multisums over hypergeometric terms to fit the input class of classic summation algorithms. Since these definite sums have nonstandard boundary conditions, the WZ-method delivers inhomogeneous recurrence relations. We designed a recursive procedure to determine the inhomogeneous parts of these recurrences and implemented it in the Mathematica package FSums which builds on the already existing packages, MultiSum and C. Schneider’s Sigma. Another approach to evaluate Feynman integrals is by representating them in terms of nested Mellin-Barnes integrals. These complex contour integrals can also be viewed as sums of residues at certain poles of the integrands and they are connected to the inversion formula for the Mellin transform. In the last part, we show how WZ-methods can be used to compute recurrences for multiple Mellin-Barnes integrals over hypergeometric terms, eliminating the need to search for sum representations. We applied this new algorithmic technique to prove typical entries from the Gradshteyn-Ryzhik table of integrals using the Mellin transform and to find recurrences for a class of Ising integrals.

Propulsion by passive filaments and active flagella near boundaries

Confinement and wall effects are known to affect the kinematics and propulsive characteristics of swimming microorganisms. When a solid body is dragged through a viscous fluid at constant velocity, the presence of a wall increases fluid drag, and thus the net force required to maintain speed has to increase. In contrast, recent optical trapping experiments have revealed that the propulsive force generated by human spermatozoa is decreased by the presence of boundaries. Here, we use a series of simple models to analytically elucidate the propulsive effects of a solid boundary on passively actuated filaments and model flagella. For passive flexible filaments actuated periodically at one end, the presence of the wall is shown to increase the propulsive forces generated by the filaments in the case of displacement-driven actuation, while it decreases the force in the case of force-driven actuation. In the case of active filaments as models for eukaryotic flagella, we demonstrate that the manner in which a solid wall affects propulsion cannot be known a priori, but is instead a nontrivial function of the flagellum frequency, wavelength, its material characteristics, the manner in which the molecular motors self-organize to produce oscillations (prescribed activity model or self-organized axonemal beating model), and the boundary conditions applied experimentally to the tethered flagellum. In particular, we show that in some cases, the increase in fluid friction induced by the wall can lead to a change in the waveform expressed by the flagella, which results in a decrease in their propulsive force.

Validation and verification issues in a timeline-based planning system

AbstractTo foster effective use of artificial intelligence planning and scheduling (P&S)systems in the real world, it is of great importance to both (a) broaden direct access to the technology for the end users and (b) significantly increase their trust in such technology. AutomatedP&Ssystems often bring solutions to the users that are neither ‘obvious’ nor immediately acceptable to them. This is because these tools directly reason on causal, temporal, and resource constraints; moreover, they employ resolution processes designed to optimize the solution with respect to non-trivial evaluation functions. Knowledge engineering environments aim at simplifying direct access to the technology for people other than the original system designers, while the integration of validation and verification (V&V) capabilities in such environments may potentially enhance the users’ trust in the technology. Somehow,V&Vtechniques may represent a complementary technology, with respect toP&S, that contributes to developing richer software environments to synthesize a new generation of robust problem-solving applications. The integration ofV&VandP&Stechniques in a knowledge engineering environment is the topic of this paper. In particular, it analyzes the use of state-of-the-artV&Vtechnology to support knowledge engineering for a timeline-based planning system called MrSPOCK. The paper presents the application domain for which the automated solver has been developed, introduces the timeline-based planning ideas, and then describes the different possibilities to applyV&Vto planning. Hence, it continues by describing the step of addingV&Vfunctionalities around the specialized planner, MrSPOCK. New functionalities have been added to perform both model validation and plan verification. Lastly, a specific section describes the benefits as well as the performance of such functionalities.

On the relation between rigging inner product and master constraint direct integral decomposition

Canonical quantization of constrained systems with first-class constraints via Dirac’s operator constraint method proceeds by the theory of Rigged Hilbert spaces, sometimes also called refined algebraic quantization. This method can work when the constraints form a Lie algebra. When the constraints only close with nontrivial structure functions, the Rigging map can no longer be defined. To overcome this obstacle, the master constraint method has been proposed which replaces the individual constraints by a weighted sum of absolute squares of the constraints. Now the direct integral decomposition (DID) methods, which are closely related to Rigged Hilbert spaces, become available and have been successfully tested in various situations. It is relatively straightforward to relate the rigging inner product to the path integral that one obtains via reduced phase space methods. However, for the master constraint, this is not at all obvious. In this paper we find sufficient conditions under which such a relation can be established. Key to our analysis is the possibility to pass to equivalent, Abelian constraints, at least locally in phase space. Then the master constraint DID for those Abelian constraints can be directly related to the rigging map and therefore has a path integral formulation.

Area distribution and the average shape of a Lévy bridge

We consider a one-dimensional Lévy bridgexB of lengthn andindex 0 < α < 2, i.e. a Lévy random walk constrained to start and end at the origin aftern time steps,xB(0) = xB(n) = 0. We compute thedistribution PB(A, n) of the area A = ∑m = 1nxB(m) under such a Lévy bridge and show that, for largen, it has the scalingform PB(A, n) ∼ n − 1 − 1/αFα(A/n1 + 1/α), with theasymptotic behavior Fα(Y) ∼ Y − 2(1 + α) for large Y. Forα = 1, we obtain an explicitexpression of F1(Y) in terms of elementary functions. We also compute the average profile at time m of a Lévybridge with fixed area A. For large n and large m and A, one finds the scaling form , where at variance with the Brownian bridge,Hα(X, Y) is a non-trivial function of the rescaled timem/n and rescaledarea Y = A/n1 + 1/α. Our analytical results are verified by numerical simulations.

Extremal memory of stochastic volatility with an application to tail shape inference

We characterize joint tails and tail dependence for a class of stochastic volatility processes. We derive the exact joint tail shape of multivariate stochastic volatility with innovations that have a regularly varying distribution tail. This is used to give four new characterizations of tail dependence. In three cases tail dependence is a non-trivial function of linear volatility memory parametrically represented by tail scales, while tail power indices do not provide any relevant dependence information. Although tail dependence is associated with linear volatility memory, tail dependence itself is nonlinear. In the fourth case a linear function of tail events and exceedances is linearly independent. Tail dependence falls in a class that implies the celebrated Hill (1975) tail index estimator is asymptotically normal, while linear independence of nonlinear tail arrays ensures the asymptotic variance is the same as the iid case. We illustrate the latter finding by simulation.

Relationships between correlation lengths and integral scales for covariance models with more than two parameters

In geostatistical applications, the terms correlation length and range are often used interchangeably and refer to a characteristic covariance length ξ that normalizes the lag distance in the variogram or the covariance model. We present equations that strictly define the correlation length (r c ) and integral range (l c ). We derive analytical expressions for r c and l c of the Whittle–Matern, fluctuation gradient curvature and rational quadratic covariances. For these covariances, we show that the correlation length and integral range for a given model are not fully determined by ξ. We define non-trivial covariance functions, and we formulate an ergodicity index based on l c . We propose using the ergodicity index to compare coarse-grained measures corresponding to non-trivial covariance functions with different parameters. Finally, we discuss potential applications of the proposed covariance models in stochastic subsurface hydrology.

Callan–Symanzik equation and asymptotic freedom in the Marr–Shimamoto model

The exactly soluble nonrelativistic Marr–Shimamoto model was introduced in 1964 as an example of the Lee model with a propagator and a nontrivial vertex function. An exactly soluble relativistic version of this model, known as the Zachariasen model, has been found to be asymptotically free in terms of coupling constant renormalization at an arbitrary spacelike momentum and on the basis of exact solutions of the Gell–Mann–Low equations. This is accomplished with conventional cut-off regularization by setting up the Yukawa and Fermi coupling constants at Euclidean momenta in terms of on mass-shell couplings and then taking the asymptotic limit. In view of this background, it may be expected that an investigation of the nonrelativistic Marr–Shimamoto theory may also exhibit asymptotic freedom in view of its manifest mathematical similarity to that of the Zachariasen model. To prove this point, the present paper prefers to examine asymptotic freedom in the nonrelativistic Marr–Shimamoto theory using the powerful concepts of the renormalization group and the Callan–Symanzik equation, in conjunction with the specificity of dimensional regularization and on-shell renormalization. This approach is based on calculations of the Callan–Symanzik coefficients and determinations of the effective coupling constants. It is shown that the Marr–Shimamoto theory is asymptotically free for dimensions D&amp;lt;3 and for values of D&amp;gt;3 occurring in periodic intervals over the range of 0&amp;lt;D&amp;lt;27 of particular interest. This differs from the original Lee model which has been shown by several authors, using these same concepts, to be asymptotically free only for D&amp;lt;4.

Bank Capital Flows and Asset Prices

We embed the notion of intermediaries as risk-reducing monitors into a dynamic general-equilibrium asset pricing framework, and consider an economy in which capital can be moved between an intermediated banking sector and a risky sector. We characterize how resources are optimally allocated between these sectors as a function of the relative size of the banking sector - the bank share - and the maximum speed at which capital can move into and out of that sector - financial flexibility. In equilibrium, the realized capital flow, i.e., the capital mobility, is a nontrivial function of the bank share and is directly related to the risk of systemic crashes. The model has strong asset pricing implications; for example, price-dividend ratios are lower the higher the financial flexibility. The relationship between financial flexibility and real growth rates is ambiguous; high financial flexibility may lead to either higher or lower growth rates. Methodologically, our paper contributes to the two-trees literature by allowing for reallocation of resources between trees in a tractable framework; this flexible-tree approach allows for stationary share distributions as an equilibrium outcome.

Linear independence of time frequency translates for special configurations

We prove that for any 4 points in the plane that belong to 2 parallel lines, there is no linear dependence between the associated time-frequency translates of any nontrivial Schwartz function. If mild Diophantine properties are satisfied, we also prove linear independence in the category of $L^2(\R)$ functions

Faster Methods for Identifying Nontrivial Energy Conservation Functions for Cellular Automata

The biggest obstacle to the efficient discovery of conserved energy functions for cellular auotmata is the elimination of the trivial functions from the solution space. Once this is accomplished, the identification of nontrivial conserved functions can be accomplished computationally through appropriate linear algebra. As a means to this end, we introduce a general theory of trivial conserved functions. We consider the existence of nontrivial additive conserved energy functions ("nontrivials") for cellular automata in any number of dimensions, with any size of neighborhood, and with any number of cell states. We give the first known basis set for all trivial conserved functions in the general case, and use this to derive a number of optimizations for reducing time and memory for the discovery of nontrivials. We report that the Game of Life has no nontrivials with energy windows of size 13 or smaller. Other $2D$ automata, however, do have nontrivials. We give the complete list of those functions for binary outer-totalistic automata with energy windows of size 9 or smaller, and discuss patterns we have observed.

Photocontrol of Gene Expressions Systems Using Light-Induced Conformational Changes of Nucleic Acids

To understand non-trivial biological functions, it is crucial to develop minimal synthetic models that capture their basic features. Here, we demonstrate a sequence-independent, reversible control of transcription and gene expression using a photosensitive nucleic acid binder (pNAB). By introducing a pNAB whose affinity for nucleic acids is tuned by light, in vitro RNA production, EGFP translation, and GFP expression were successfully inhibited in the dark and recovered after a short illumination at 365 nm. Our results indicate that the accessibility of the protein machinery to one or several nucleic acid binding sites can be efficiently regulated by changing the conformational/condensation state of the nucleic acid (DNA conformation or mRNA aggregation), thus regulating gene activity in an efficient, reversible, and sequence-independent manner. The possibility offered by our approach to use light to trigger various gene expression systems in a system-independent way opens interesting perspectives to study gene expression dynamics as well as to develop photocontrolled biotechnological procedures.Ref: A. Estevez-Torres, C. Crozatier, A. Diguet, T. Hara, H. Saito, K. Yoshikawa, D. Baigl, Proc. Natl. Acad. Sci. USA 2009, 106, 12219-12223.View Large Image | View Hi-Res Image | Download PowerPoint Slide

On a Ströbel-Inspired k(t) FLRW Ansatz in a Class of Metric F(R) Models

Assuming a D≥4 dimensional FLRW (Friedmann–Lemaitre–Robertson–Walker) inspired ansatz with spatial curvature being a non-trivial function of time k(t) in a class of metric and Jordan frame F(R) gravity models, non-existence theorems for several types of sources are derived in a simple manner (using specific form of the modified gravity Einstein tensor components).

Dark solitons in holographic superfluids

We construct dark soliton solutions in a holographic model of a relativistic superfluid. We study the length scales associated with the condensate and the charge density depletion, and find that the two scales differ by a non-trivial function of the chemical potential. By adjusting the chemical potential, we study the variation of the depletion of charge density at the interface.

Sequence-independent and reversible photocontrol of transcription/expression systems using a photosensitive nucleic acid binder.

To understand non-trivial biological functions, it is crucial to develop minimal synthetic models that capture their basic features. Here, we demonstrate a sequence-independent, reversible control of transcription and gene expression using a photosensitive nucleic acid binder (pNAB). By introducing a pNAB whose affinity for nucleic acids is tuned by light, in vitro RNA production, EGFP translation, and GFP expression (a set of reactions including both transcription and translation) were successfully inhibited in the dark and recovered after a short illumination at 365 nm. Our results indicate that the accessibility of the protein machinery to one or several nucleic acid binding sites can be efficiently regulated by changing the conformational/condensation state of the nucleic acid (DNA conformation or mRNA aggregation), thus regulating gene activity in an efficient, reversible, and sequence-independent manner. The possibility offered by our approach to use light to trigger various gene expression systems in a system-independent way opens interesting perspectives to study gene expression dynamics as well as to develop photocontrolled biotechnological procedures.

Asymptotic behavior of solutions for a Lotka–Volterra mutualism reaction–diffusion system with time delays

This paper is to investigate the asymptotic behavior of solutions for a time-delayed Lotka–Volterra N -species mutualism reaction–diffusion system with homogeneous Neumann boundary condition. It is shown, under a simple condition on the reaction rates, that the system has a unique bounded time-dependent solution and a unique constant positive steady-state solution, and for any nontrivial nonnegative initial function the corresponding time-dependent solution converges to the constant positive steady-state solution as time tends to infinity. This convergence result implies that the trivial steady-state solution and all forms of semitrivial steady-state solutions are unstable, and moreover, the system has no nonconstant positive steady-state solution. A condition ensuring the convergence of the time-dependent solution to one of nonnegative semitrivial steady-state solutions is also given.

Localized magnetic states in Rashba dots

We study the formation of local moments in quantum dots arising in quasi-one-dimensional electron wires due to localized spin-orbit (Rashba) interaction. Using an Anderson-type model to describe the occurrence of the magnetic moments in these Rashba dots, we calculate the local magnetization within the mean-field approximation. We find that the magnetization becomes a nontrivial function of the Rashba coupling strength. We discuss both the equilibrium and nonequilibrium cases. Interestingly, we obtain a magnetic phase which is stable at large bias due to the Rashba interaction.

Nontangential limits in 𝒫t(μ)-spaces and the index of invariant subgroups

Let mu be a finite positive measure on the closed disk (D) over bar in the complex plane, let 1 <= t < infinity, and let P-t(mu) denote the closure of the analytic polynomials in L-t(mu). We suppose that D is the set of analytic bounded point evaluations for P-t(mu), and that P-t(mu) contains no nontrivial characteristic functions. It is then known that the restriction of mu to partial derivative D must be of the form h vertical bar dz vertical bar. We prove that every function f is an element of P-t(mu) has nontangential limits at h vertical bar dz vertical bar-almost every point of partial derivative D, and the resulting boundary function agrees with f as an element of L-t(h vertical bar dz vertical bar). Our proof combines methods from James E. Thomson's proof of the existence of bounded point evaluations for P-t(mu) whenever P-t(mu) not equal L-t(mu) with Xavier Tolsa's remarkable recent results on analytic capacity. These methods allow us to refine Thomson's results somewhat. In fact, for a general compactly supported measure nu in the complex plane we are able to describe locations of bounded point evaluations for P-t(nu) in terms of the Cauchy transform of an annihilating measure. As a consequence of our result we answer in the affirmative a conjecture of Conway and Yang. We show that for 1 < t < infinity dim M/zM = 1 for every nonzero invariant subspace M of P-t(mu) if and only if h not equal 0. We also investigate the boundary behaviour of the functions in P-t(mu) near the points z is an element of partial derivative D where h(z) = 0. In particular, for 1 < t < infinity we show that there are interpolating sequences for P-t(mu) that accumulate nontangentially almost everywhere on {z : h(z) = 0}. (Less)