Variational Formula Research Articles

Study on Sustainable Well Water Pumping Technology to Melt Ice in Winter for the Middle Route of the South-to-North Water Diversion Project

In order to improve the water delivery capacity of the middle route of the South-to-North Water Diversion Project in winter, the technology of pumping well water to melt ice was previously adopted to improve the water temperature of the channel. In order to protect the local ecology and channel water quality during the process of pumping and recharging, this paper has analyzed the optimal limit to the range of groundwater level fluctuations and groundwater temperature fluctuations. The Baiquan Underground Reservoir was considered as the research object, and a three-dimensional numerical model of hydrothermal coupling was established. Correlation analyses of pumping and recharge capacity with various factors, and between temperature changes after recharging and various factors, were carried out with a view to protecting local water quality. The single well pumping and recharging scheme was optimized by genetic algorithm. The quantitative estimation formula of pumping and recharge water capacity, and the well depth, permeability coefficient, initial water level, and duration of continuous pumping and recharge, as well as the recovery water level variation formula, were established under the limitation of water level fluctuation. The quantitative relationship between horizontal maximum diameter, L, and the recharge flow rate, the continuous recharge duration, vertical maximum depth, H; and the well depth, temperature change rate, K, and the recharge water temperature, were obtained. The results showed that the permeability coefficient and soil thermal conductivity had little effect on the temperature field change. Additionally, recharge should be carried out in summer, when the temperature of recharge water is above 14.8 °C. When the number of pumping or recharging times, n, is two, the total amount of pumping or recharging during the operation period reaches its maximum. Compared with continuous operation during the operating period, this report found that the total amount of pumped water would increase by 1.1 × 106 m3, and the total amount of recharge would increase by 7 × 105 m3.

The Parabolic Anderson Model on a Galton–Watson Tree Revisited

In den Hollander et al. (The parabolic Anderson model on a Galton-Watson tree, to appear in in and out of equilibrium 3: celebrating Vladas Sidoravicius. Progress in probability, Birkhäuser, Basel, 2021) a detailed analysis was given of the large-time asymptotics of the total mass of the solution to the parabolic Anderson model on a supercritical Galton–Watson random tree with an i.i.d. random potential whose marginal distribution is double-exponential. Under the assumption that the degree distribution has bounded support, two terms in the asymptotic expansion were identified under the quenched law, i.e., conditional on the realisation of the random tree and the random potential. The second term contains a variational formula indicating that the solution concentrates on a subtree with minimal degree according to a computable profile. The present paper extends the analysis to degree distributions with unbounded support. We identify the weakest condition on the tail of the degree distribution under which the arguments in den Hollander et al. (The parabolic Anderson model on a Galton-Watson tree, to appear in in and out of equilibrium 3: celebrating Vladas Sidoravicius. Progress in probability, Birkhäuser, Basel, 2021) can be pushed through. To do so we need to control the occurrence of large degrees uniformly in large subtrees of the Galton–Watson tree.

Finite element analysis and shape optimization of singular points in boundary value problems for partial differential equations

We introduce the concept of shape optimization of singular points of a weak solution in boundary value problems of partial differential equations (BVP) including fracture mechanics, shape optimization of boundary and interface, etc. Roughly speaking, singularity is a gap between weak and strong solutions. We also introduce the integral formula named the GJ-integral defined with an arbitrary subdomain as an extension of the J-integral in fracture mechanics. The GJ-integral is defined locally, is made from partial differential equations in BVP only, takes zero if the solution is regular, and expresses energy shape sensitivity. A generalization of the Hadamard variational formula is proposed and various cost functions are rewritten using the GJ-integral. In the last section there are numerical calculations using the GJ-integral with the finite element method.

Corner Detection of the Computer VR Microscope Image Based on the 3D Reconstruction Algorithm.

In order to solve the problem of multisolution and ill-formedness of the 3D reconstruction method of a single image (purpose), the author proposes a microscope image segmentation algorithm based on the Harris multiscale corner detection. Separating complex engineering images into several simple basic geometric shapes, rebuild them separately to avoid the ill-conditioned solution problem of directly recovering depth information. In order to improve the registration accuracy of the corner-based image registration algorithm, the idea of multiresolution analysis was introduced into the classic Harris corner detection, and a gray intensity variation formula based on the wavelet transform was constructed, and a scale transformation characteristic was obtained so that the improved Harris corner detection algorithm is invariant to rotation, translation, and scale. Experimental results show that after reconstruction, the error between the length of the object measured based on the point cloud data and the actual length of the object is small, and both remain within the error range of 3 mm. The experiment verifies the fast, accurate, and stable characteristics of the improved algorithm.

Dynamics for a two-phase free boundary system in an epidemiological model with couple nonlocal dispersals

The present paper is devoted to the investigation of the long time dynamics for a double free boundary system with nonlocal diffusions, which models the infectious diseases transmitted via digestive system such as fecal-oral diseases, cholera, hand-foot and mouth, etc... We start by proving the existence and uniqueness of the Cauchy problem, which is not a trivial step due to presence of couple nonlocal dispersals and new types of nonlinear reaction terms. Next, we provide simple conditions on comparing the basic reproduction numbers R0 and R⁎ with 1 to characterize the global dynamics, as t→∞. We further obtain the sharp criteria for the spreading and vanishing in term of the initial data. This is also called the vanishing-spreading phenomena. The couple dispersals yield significant obstacle that we cannot employ the approach of Zhao, Zhang, Li, Du [34] and Du-Ni [14]. To overcome this, we must prove the existence and the variational formula for the principal eigenvalue of a linear system with nonlocal dispersals, then use it to obtain the right limits as the dispersal rates and domain tend to zero or infinity. The maximum principle and sliding method for the nonlocal operator are ingeniously employed to achieve the desired results.

Resolvent expansions for self‐adjoint operators via boundary triplets

In this paper, we develop certain aspects of perturbation theory for self-adjoint operators subject to small variations of their domains. We use the abstract theory of boundary triplets to quantify such perturbations and give the second-order asymptotic analysis for resolvents, spectral projections, and discrete eigenvalues of the corresponding self-adjoint operators. In particular, we derive explicit formulas for the first variation and the Hessian of the eigenvalue curves bifurcating from a discrete eigenvalue of an unperturbed operator. An application is given to a matrix valued Robin Laplacian and more general Robin-type self-adjoint extensions.

Quasi-invariant Gaussian measures for the nonlinear wave equation in three dimensions

We prove quasi-invariance of Gaussian measures supported on Sobolev spaces under the dynamics of the three-dimensional defocusing cubic nonlinear wave equation. As in the previous work on the two-dimensional case, we employ a simultaneous renormalization on the energy functional and its time derivative. Two new ingredients in the three-dimensional case are (i) the construction of the weighted Gaussian measures, based on a variational formula for the partition function inspired by Barashkov and Gubinelli (2018), and (ii) an improved argument in controlling the growth of the truncated weighted Gaussian measures, where we combine a deterministic growth bound of solutions with stochastic estimates on random distributions.

Optimizing Variational Representations of Divergences and Accelerating Their Statistical Estimation

Variational representations of divergences and distances between high-dimensional probability distributions offer significant theoretical insights and practical advantages in numerous research areas. Recently, they have gained popularity in machine learning as a tractable and scalable approach for training probabilistic models and for statistically differentiating between data distributions. Their advantages include: 1) They can be estimated from data as statistical averages. 2) Such representations can leverage the ability of neural networks to efficiently approximate optimal solutions in function spaces. However, a systematic and practical approach to improving the tightness of such variational formulas, and accordingly accelerate statistical learning and estimation from data, is currently lacking. Here we develop such a methodology for building new, tighter variational representations of divergences. Our approach relies on improved objective functionals constructed via an auxiliary optimization problem. Furthermore, the calculation of the functional Hessian of objective functionals unveils the local curvature differences around the common optimal variational solution; this quantifies and orders the tightness gains between different variational representations. Finally, numerical simulations utilizing neural network optimization demonstrate that tighter representations can result in significantly faster learning and more accurate estimation of divergences in both synthetic and real datasets (of more than 1000 dimensions), often accelerated by nearly an order of magnitude.

Crisanti–Sommers Formula and Simultaneous Symmetry Breaking in Multi-species Spherical Spin Glasses

There is a rich history of expressing the limiting free energy of mean-field spin glasses as a variational formula over probability measures on $[0,1]$, where the measure represents the similarity (or "overlap") of two independently sampled spin configurations. At high temperatures, the formula's minimum is achieved at a measure which is a point mass, meaning sample configurations are asymptotically orthogonal up to a magnetic field correction. At low temperatures, though, a very different behavior emerges known as replica symmetry breaking (RSB). The deep wells in the energy landscape create more rigid structure, and the optimal overlap measure is no longer a point mass. The exact size of its support remains in many cases an open problem. Here we consider these themes for multi-species spherical spin glasses. Following a companion work in which we establish the Parisi variational formula, here we present this formula's Crisanti-Sommers representation. In the process, we gain new access to a problem unique to the multi-species setting. Namely, if RSB occurs for one species, does it necessarily occur for other species as well? We provide sufficient conditions for the answer to be yes. For instance, we show that if two species share any quadratic interaction, then RSB for one implies RSB for the other. Moreover, the level of symmetry breaking must be identical, even in cases of full RSB. In the presence of an external field, any type of interaction suffices.

Bismut–Cheeger Eta Form and Higher Spectral Flow

Abstract We prove the variation formula, the embedding formula, and the adiabatic limit formula for equivariant Bismut–Cheeger eta forms with a fiberwise compact Lie group action. We only need the condition that the kernels of corresponding fiberwise Dirac operators form vector bundles.

Variational inference of the drift function for stochastic differential equations driven by Lévy processes.

In this work, we consider the nonparametric estimation problem of the drift function of stochastic differential equations driven by the α-stable Lévy process. We first optimize the Kullback-Leibler divergence between the path probabilities of two stochastic differential equations with different drift functions. We then construct the variational formula based on the stationary Fokker-Planck equation using the Lagrangian multiplier. Moreover, we apply the empirical distribution to replace the stationary density, combining it with the data information, and we present the estimator of the drift function from the perspective of the process. In the numerical experiment, we investigate the effect of the different amounts of data and different α values. The experimental results demonstrate that the estimation result of the drift function is related to both and that the exact drift function agrees well with the estimated result. The estimation result will be better when the amount of data increases, and the estimation result is also better when the α value increases.

Effectiveness of Soil Erosion Monitoring Strategies in Kuresoi South, Kenya

Soil erosion is still one of the most important land problems and it is linked to land use and land cover changes. To mitigate against this, there has been an effort to transfer the management of natural resources to local communities especially in the tropics. Many of these initiatives incorporate the development of monitoring systems. Local monitoring systems constitute an almost compulsory component of any program or project dealing with sustainable management of natural resources. The purpose of this study was to assess soil erosion monitoring indicators as effective management tools to be used to help in monitoring change in natural resources within the study area. The study was comprised of a total representative sample population of 68 respondents from Kuresoi south catchment which was achieved using Nassiuma coefficient of variation formulae. The study adopted a descriptive research design. Data and information was collected using questionnaires and interview schedules. Both quantitative and qualitative data was analyzed using Statistical Package for Social Sciences and presented in form of charts, tables of frequencies and percentages. The study results revealed the effectiveness of soil erosion monitoring indicators, degree of soil erosion and natural resource management. The study therefore recommends that it is important to introduce capacity building programs to the community and other stakeholders through creation of awareness and training so that they can have knowledge on these factors and thus implement the appropriate mitigation measures.

Willmore surfaces and Hopf tori in homogeneous 3-manifolds

Some classification results for closed surfaces in Berger spheres are presented. On the one hand, a Willmore functional for isometrically immersed surfaces into an homogeneous space {mathbb {E}}^{3}(kappa ,tau ) with isometry group of dimension 4 is defined and its first variational formula is computed. Then, we characterize Clifford and Hopf tori as the only Willmore surfaces satisfying a sharp Simons-type integral inequality. On the other hand, we also obtain some integral inequalities for closed surfaces with constant extrinsic curvature in {mathbb {E}}^3(kappa ,tau ), becoming equalities if and only if the surface is a Hopf torus in a Berger sphere.

A Remark on the Spherical Bipartite Spin Glass

Auffinger and Chen (J Stat Phys 157:40–59, 2014) proved a variational formula for the free energy of the spherical bipartite spin glass in terms of a global minimum over the overlaps. We show that a different optimisation procedure leads to a saddle point, similar to the one achieved for models on the vertices of the hypercube.

Theory of Parcel Vorticity Evolution in Supercell-Like Flows

Abstract In a prior paper, insights into tornadogenesis in supercell storms were gained by discovering analytical formulas for vorticity variations along streamlines in idealized, steady, frictionless, isentropic inflows of dry air imported from a horizontally uniform environment. This work is simplified and extended to the evolution of parcel vorticity in unsteady, nonisentropic flows by integrating the vorticity equation using nonorthogonal Lagrangian coordinates. The covariant basis vectors , , and are material line elements attached to each parcel. Initially they form an orthonormal set with in the direction of and left normal to the storm-relative wind at each level in the environment, and upward. The surface containing all parcels with the same initial height constitutes a material surface, within which initially streamwise and transverse material lines are reoriented and stretched or shrunk. The basis vectors propagate a parcel’s barotropic vorticity through time by factoring in the “frozen-field” effect. With a horizontally uniform environment, the barotropic vorticity of a parcel depends on its initial streamwise vorticity times its current plus its initial crosswise vorticity times its current . For baroclinic and frictional vorticity, each contravariant component is the integral from initial to current time of the corresponding contravariant component of the generation vector. The “river-bend” effect acting on all parts (baroclinic, frictional, and barotropic) of transverse vorticity produces streamwise vorticity (parallel to 3D wind). In left-turning steady flow, it arises from rotating toward . For steady, frictionless, dry isentropic flow, previous vorticity formulas are recovered. Significance Statement Air parcels rising in a tornado spin rapidly about their direction of motion. Theory herein describes the processes that can produce this streamwise spin in supercells. Cyclonic updraft rotation originates from strong low-level environmental storm-relative winds that turn clockwise with height. Parcels flowing into the updraft have initially large streamwise spins that are amplified by streamwise stretching. Rain curtains falling through the cyclonic updraft cause other parcels to descend and turn leftward. Buoyancy and frictional torques give them horizontal spin. Even if these spins are transverse to the flow initially, they are turned streamwise by secondary flow that develops in left-hand bends. As the parcels reach the ground and converge into the tornado, streamwise stretching greatly magnifies their streamwise spins.

A shape theorem and a variational formula for the quenched Lyapunov exponent of random walk in a random potential

Nous prouvons un théorème de forme et déduisons une formule variationnelle pour l’exposant de Lyapunov et la fonction de Green de la marche aléatoire dans un potentiel aléatoire sur un réseau carré de dimension arbitraire et avec un ensemble fini arbitraire des pas possibles. Le potentiel est une fonction d’un environnement stationnaire et du pas de la marche. Ce potentiel est soumis à une hypothèse sur les moments qui est liée à la vitesse de mélange du milieu. Notre cadre comprend les modèles de polymères dirigés et non dirigés, les marche aléatoire dans un environnement aléatoire statique et dynamique, et, dans le cas de température nulle, nos résultats donnent également un théorème de forme et une formule variationnelle pour la constante de temps de la percolation du dernier passage dirigée par site et par arête et de la percolation du premier passage standard.

Variational formulas for curves of fixed degree

We consider a length functional for $C^1$ curves of fixed degree in graded manifolds equipped with a Riemannian metric. The first variation of this length functional can be computed only if the curve can be deformed in a suitable sense, and this condition is expressed via a differential equation along the curve. In the classical differential geometry setting, the analogous condition was considered by Bryant and Hsu in [7, 21] who proved that it is equivalent to the surjectivity of a holonomy map. The purpose of this paper is to extend this deformation theory to curves of fixed degree providing several examples and applications. In particular, we give a useful sufficient condition to guarantee the possibility of deforming a curve.

A Steiner Inequality for the Anisotropic Perimeter

In this paper, we prove the monotonicity of the anisotropic perimeter of sets of finite perimeter under Steiner symmetrization by a variational formula of volume and an inequality for the anisotropic lower outer Minkowski content. As a consequence, we give a more direct proof of the Wulff inequality by Steiner symmetrization.

A Novel Second-Order and Unconditionally Energy Stable Numerical Scheme for Allen–Cahn Equation

We propose a novel time-stepping scheme for solving the Allen–Cahn equation. We first rewrite the free energy into an equivalent form and then obtain a new Allen–Cahn equation by energy variational formula of L 2 -gradient flow. Using leapfrog formula, a new linear scheme is obtained, and we prove that the numerical scheme is unconditionally energy stable and uniquely solvable, and the discrete energy is in agreement with the original free energy. In addition, we also discuss the uniform boundedness and error estimate of numerical solution, the results show that the numerical solution is uniformly bounded in H 2 -norm, and error estimate shows that the time-stepping scheme can achieve second-order accuracy in time direction. At last, several numerical tests are illustrated to verify the theoretical results. The numerical strategy developed in this paper can be easily applied to other gradient flow models.

A generalization of multiple zeta values. Part 2: Multiple sums

{\bf Abstract:} Multiple zeta values have become of great interest due to their numerous applications in mathematics and physics. In this article, we present a generalization, which we will refer to as multiple sums, where the reciprocals are replaced with arbitrary sequences. We develop formulae to help with manipulating such sums. We develop variation formulae that express the variation of multiple sums in terms of lower order multiple sums. Additionally, we derive a set of partition identities that we use to prove a reduction theorem that expresses multiple sums as a combination of simple sums. We present a variety of applications including applications concerning polynomials and MZVs such as generating functions and expressions for $\zeta(\{2p\}_m)$ and $\zeta^\star(\{2p\}_m)$. Finally, we establish the connection between multiple sums and a type of sums called recurrent sums. By exploiting this connection, we provide additional partition identities for odd and even partitions.