Variational Formula Research Articles

Wave Propagation for Reaction-Diffusion Equations on Infinite Random Trees

The asymptotic wave speed for FKPP type reaction-diffusion equations on a class of infinite random metric trees are considered. We show that a travelling wavefront emerges, provided that the reaction rate is large enough. The wavefront travels at a speed that can be quantified via a variational formula involving the random branching degrees $\vec{d}$ and the random branch lengths $\vec{\ell}$ of the tree. This speed is slower than that of the same equation on the real line $\mathbb{R}$, and we estimate this slow down in terms of $\vec{d}$ and $\vec{\ell}$. Our key idea is to project the Brownian motion on the tree onto a one-dimensional axis along the direction of the wave propagation. The projected process is a multi-skewed Brownian motion, introduced by Ramirez [Multi-skewed Brownian motion and diffusion in layered media, Proc. Am. Math. Soc., Vol. 139, No. 10, pp.3739-3752, 2011], with skewness and interface sets that encode the metric structure $(\vec{d}, \vec{\ell})$ of the tree. Combined with analytic arguments based on the Feynman-Kac formula, this idea connects our analysis of the wavefront propagation to the large deviations principle (LDP) of the multi-skewed Brownian motion with random skewness and random interface set. Our LDP analysis involves delicate estimates for an infinite product of $2\times 2$ random matrices parametrized by $\vec{d}$ and $\vec{\ell}$ and for hitting times of a random walk in random environment. By exhausting all possible shapes of the LDP rate function (action functional), the analytic arguments that bridge the LDP and the wave propagation overcome the random drift effect due to multi-skewness.

A variational characterization of the optimal exit rate for controlled diffusions

The main result in this paper is a variational formula for the exit rate from a bounded domain for a diffusion process in terms of the stationary law of the diffusion constrained to remain in this domain forever. Related results on the geometric ergodicity of the controlled Q-process are also presented.

Dimensions of random statistically self-affine Sierpinski sponges in [formula omitted

We compute the Hausdorff dimension of any random statistically self-affine Sierpinski sponge K⊂Rk (k≥2) obtained by using some percolation process in [0,1]k. To do so, we first exhibit a Ledrappier-Young type formula for the Hausdorff dimensions of statistically self-affine measures supported on K. This formula presents a new feature compared to its deterministic or random dynamical version. Then, we establish a variational principle expressing dimH⁡K as the supremum of the Hausdorff dimensions of statistically self-affine measures supported on K, and show that the supremum is uniquely attained. The value of dimH⁡K is also expressed in terms of the weighted pressure function of some deterministic potential. As a by-product, when k=2, we give an alternative approach to the Hausdorff dimension of K, which was first obtained by Gatzouras and Lalley [27]. The value of the box counting dimension of K and its equality with dimH⁡K are also studied. We also obtain a variational formula for the Hausdorff dimensions of some orthogonal projections of K, and for statistically self-affine measures supported on K, we establish a dimension conservation property through these projections.

Explaining the Nature of the Mass m of Submicroparticles and the Phenomenon of Mass Variation with Velocity V in Ether

A brief history of mass nature is presented, including formulas for mass variation with velocity V. A new mass hypothesis (MH), on the nature of mass m is presented. Accordingly, the mass m0 of submicroparticle (SMP), is given by the kinetic energy accumulated inside the SMP, by the vortices/vibrations of groups of ether (ETH) cells, constituting the submicroparticles (SSMP). These SSMPs will also create the inertial force (FI) of the SMP, due to the opposition manifested by ETH, toward any accelerated movement of SMP. A new explanation of mass m increase, with V is proposed, admitting the existence of ether in the form of our model, HM16. The entrainment of the surrounding ETH of SMP, takes place, with speed VETH, which stores the EETH energy. The spatial variation of VETH around the SMP, was analysed using the method of potential motions, for a cylinder, moving in water-like fluid, or in ETH, resulting a spectrum of the lines of current ψ, indicating the size of VETH, rapidly decreasing with distance r. So resulted the kinetic energy EETH stored by the movement of the outer ETH, which provides the supplementary mass ms. The case of SMP which acquires velocity V, approaching c, was analysed. Then, a phase change (PC) will take place, through which the SMP, transforms into photons (PHs), when movements of the ETH from the outside including energy EETH is transferred into PH photons, having velocity c, representing ETH’s property.

On thermal stability of piezo-flexomagnetic microbeams considering different temperature distributions

By relying on the Euler–Bernoulli beam model and energy variational formula, we indicate critical temperature causes in the buckling of piezo-flexomagnetic microscale beams. The corresponding size-dependent approach is underlying as a second strain gradient theory. Small deformations of elastic solids are assessed, and the mathematical discussion is linear. Regardless of the pyromagnetic effects, the thermal loading of the thermal environment varies in three states along with the thickness, which is linear, uniform, and parabolic forms. We then establish the results by developing consistent shape functions that independently evaluate boundary conditions. Next, we analytically develop and explore the effective properties of the studied beam concerning vital factors. It was achieved that piezomagnetic-flexomagnetic microbeams are more affected by the thermal environment while the thermal loading is parabolically distributed across the thickness, particularly when the boundaries involve simple supports.

Optimal Markovian Coupling and Exponential Convergence Rate for the TCP Process

In this work, by constructing optimal Markovian couplings we investigate exponential convergence rate in the Wasserstein distance for the transmission control protocol process. Most importantly, we provide a variational formula for the lower bound of the exponential convergence rate.

Python Language Training System Based on MFCC, VQ, Variational Coefficient and KNTM Algorithm

This contribution describes the second stage of the creation of a language training system programmed in Python with the aim of application to speech therapy in spanish-speaking countries, starting the study in Cuba. The first stage of this research was carried out in Matlab by analyzing the dynamics of change of the centroids of the codebooks, extracted from words pronounced by a locutor. As second stage, the Variational Coefficient formula is used in order to estimate the percentage of effectiveness with which the announcer performs voice training. A modified approach to programming the variational coefficient is taken into account as a measure of dispersion of a group of vectors. The modification is given by taking the mean of the group of vectors as the vector that represents the phonetic boundaries of the word to be trained. Besides, a novel approach for word recognition is used, based on the K-Nearest Training Matrix (KNTM) algorithm that lays its foundations in the analysis of matrix similarity taken the Frobenius norm as a measure to distinguish similar or non-similar characteristics of a matrix with respect to a database of matrices. To reduce the computational cost of the program and speed up its proper functioning, the training matrices of the database are saved in files with a .tex extension, in this way after training process, the program should only read them and not recalculate them, which significantly reduces the running time of the algorithm.

STOCHASTIC DIFFERENTIAL GAMES IN DISTRIBUTED SYSTEMS WITH DELAY

We study a differential game of approach in a delay stochastic system. The evolution of the system is described by Ito`s linear stochastic differential equation in Hilbert space. The considered Hilbert spaces are assumed to be real and separable. The Wiener process takes values in a Hilbert space and has a nuclear symmetric positive covariance operator. The pursuer and evader controls are non-anticipating random processes, taking on values, generally, in different Hilbert spaces. The operator multiplying the system state is the generator of an analytic semigroup. Solutions of the equation are represented with the help of a formula of variation of constants by the initial data and the control block. The delay effect is taken into account by summing shift type operators. To study the differential game, the method of resolving functions is extended to case of delay stochastic systems in Hilbert spaces. The technique of set-valued mappings and their selectors is used. We consider the application of obtained results in abstract Hilbert spaces to systems described by stochastic partial differential equations with time delay. By taking into account a random external influence and time delay, we study the heat propagation process with controlled distributed heat source and leak.

Applied Physics and Technologies in Football, The Earth´s Gravitational Field and Gravitational Energy Emission, Effects of the velocity, air viscosity (Bernoulli Effect) and turbulence (Magnus Effect) on the Football ball and Effects of the Air Resistance, Atmospheric Pressure and Gravitation in Football and Players: The Variant Pressure and Variant Gravity respect to the altitude of the place above sea level:

It is possible to play Football on Earth because of Gravity and almost stable and suitable atmospheric conditions6,7,8. It is demonstrated at this article real gravitational interaction between objects with mass with Cavendish Experiment4 which is described by means of gravitational field g through a distance action interaction2,3,6. The gravitational energy emission and effect of gravitation as gravitational redshift, deflection of light passing close to Sun4 and Perihelion Precession of Mercury4 with discovered formulas obtained by myself are demonstrated at article4: “The Fundament of Mass and Effects of Gravitation on a Particle and Light”4. They are in agreement with experiment results and with Theory of General Relativity with high accuracy4. In addition, electromagnetic energy emission (photons) with corresponding decrease in mass for electron (Maxwell´s theory) with discovered formula obtained by myself is demonstrated at research article of Variant Mass for an accelerated charged particle5. Relating Football, it is investigated related applied Physics to Football as for example Free fall, Parabolic movement, Newton Laws, Work and Energy, Principle of energy conservation, Impulse and Momentum and Collisions1,2,3. Besides, it is researched effects of air resistance in maximum distance and height reached by ball in parabolic movement in a Free kick for example. However, main objective of this article is to research about effects of variable physics as air resistance, atmospheric pressure and gravity and climatic conditions (heavy rain, wind, snow, hail, high temperature or humidity) that influence in match and in Football ball: the ball does not bend and in Football players (effects of altitude in organism)6. It is obtained and demonstrated at this article formulas of variation of atmospheric pressure and gravity respect to altitude of place or city above sea level. For other hand, it is researched effects of air viscosity (Bernoulli Effect) and turbulence (Magnus Effect) on Football ball when player kicks with high velocity. Besides, it is researched effects of variations of atmospheric pressure and gravity in Football players and Physics of devices to measure atmospheric pressure and blood pressure which is important when they plays a Football match in a high altitude city or place6,7,8. It is also researched effects of climatic conditions in a Football match. Even weather conditions (high altitude, warm climates) and location can be used in favor to make result favorable. For other hand, weather conditions are very important for development of a good Football game. Many times matches are suspended due extreme climatic conditions as heavy rain, wind, snow, hail or severe weather conditions as high temperature or humidity. However, it has been possible to apply technology to build new stadiums that are suitable for these climates and to wear respective clothing that facilitates practice of Football6,7,8. Therefore, it is also researched at this article new technology that can be applied for normal development of a Football game and to get efficient performance of Football players and all best physical conditions on Football field 6. Thus, technologies have been incorporated to know characteristics of players that are very helpful to coach and physical trainer to obtain efficiency of player and of strategies applied in Football field. It is very important to use all advantages, characteristics and resources of team at appropriate time6. In this way, it is maximized important details that can contribute to win Football match. It helps a lot technical body such as Coach, Technical assistant, Physical trainer, Doctor, Managers, Scientists, who know very well all best that team has and physical conditions, strategies and player skills that can be used in a certain Football game. Thus, Teamwork is key to this beautiful sport: The Football6,7,8. Moreover, Medical Technologies have been incorporated in Football6,7,8 and in future, new technologies can also be incorporated to monitor player performance.

Flats, spikes and crevices: the evolving shape of the inhomogeneous corner growth model

We study the macroscopic evolution of the growing cluster in the exactly solvable corner growth model with independent exponentially distributed waiting times. The rates of the exponentials are given by an addivitely separable function of the site coordinates. When computing the growth process (last-passage times) at each site, the horizontal and vertical additive components of the rates are allowed to also vary respectively with the column and row number of that site. This setting includes several models of interest from the literature as special cases. Our main result provides simple explicit variational formulas for the a.s. first-order asymptotics of the growth process under a decay condition on the rates. Subject to further mild conditions, we prove the existence of the limit shape and describe it explicitly. We observe that the boundary of the limit shape can develop flat segments adjacent to the axes and spikes along the axes. Furthermore, we record the formation of persistent macroscopic spikes and crevices in the cluster that are nonetheless not visible in the limit shape. As an application of the results for the growth process, we compute the flux function and limiting particle profile for the TASEP with the step initial condition and disorder in the jump rates of particles and holes. Our methodology is based on concentration bounds and estimating the boundary exit probabilities of the geodesics in the increment-stationary version of the model, with the only input from integrable probability being the distributional invariance of the last-passage times under permutations of columns and rows.

Impulsive Effects Based Distributed Synchronization of Heterogeneous Coupled Neural Networks

This paper is devoted to investigating the exponential synchronization problem on a class of coupled heterogeneous neural networks with hybrid time-varying delays. Since the heterogeneity of neural networks, the quasi-synchronization rather than complete synchronization is studied. Consider that the controller would suffer from impulsive disturbances, the distributed pinning control strategy is introduced. By jointly applying the concept of average impulsive intervals, the extended comparison principle of impulsive systems, and the vector norm techniques, sufficient conditions for the achievement of exponential quasi-synchronization on coupled heterogeneous neural networks are eventually obtained. In view that different impulsive effects play different roles in network synchronizing, two different situations of exponential synchronization are considered, respectively, by separately discussing the impulsive-effects-related parameter. Additionally, on account of the extended parameter variation formula for impulsive systems with hybrid time-varying delays, the exponential convergence velocities and the quasi-synchronization errors are estimated, separately, with respect to different ranges of impulsive effects. At last, three numerical examples with distinct impulsive effect values are given to illustrate the effectiveness of the theoretical analysis and the control schemes.

Parameters Variation-Based Synchronization on Derivative Coupled Lur’e Networks

This paper investigates the exponential synchronization of coupled Lur’e dynamical networks with multiple time-varying delays and derivative coupling. In order to synchronize the Lur’e dynamical networks to the corresponding Lur’e systems, we propose a kind of impulsive pinning control strategy, where different functions of impulsive effects are taken into account. Sufficient conditions are derived for the exponential synchronization of the derivative coupled Lur’e dynamical networks by jointly applying the contradiction proof method, the concept of an average impulsive interval, comparison principle, and the extended parameters variation formula. Simultaneously, the convergence rates of exponential synchronization are obtained according to the definition of the impulsive solution equation on different functions of impulsive effects. Furthermore, three numerical examples are presented to demonstrate the validity of the theoretical analysis and the control protocol.

Decomposition of generalized O’Hara’s energies

O’Hara introduced several functionals as knot energies. One of them is the Mobius energy. We know its Mobius invariance from Doyle-Schramm’s cosine formula. It is also known that the Mobius energy was decomposed into three components keeping the Mobius invariance. The first component of decomposition represents the extent of bending of the curves or knots, while the second one indicates the extent of twisting. The third one is an absolute constant. In this paper, we show a similar decomposition for generalized O’Hara energies. We also extend the cosine formula for the Mobius energy to generalized O’Hara energies. It gives us a condition for which the right circle minimizes the energy under the length-constraint. Furthermore, it shows us how far the energy is from the Mobius invariant property. Using decomposition, the first and second variation formulae are derived.

Propagation phenomena for a criss-cross infection model with non-diffusive susceptible population in periodic media

<p style='text-indent:20px;'>This paper is concerned with propagation phenomena for an epidemic model describing the circulation of a disease within two populations or two subgroups in periodic media, where the susceptible individuals are assumed to be motionless. The spatial dynamics for the cooperative system obtained by a classical transformation are investigated, including spatially periodic steady state, spreading speeds and pulsating travelling fronts. It is proved that the minimal wave speed is linearly determined and given by a variational formula involving linear eigenvalue problem. Further, we prove that the existence and non-existence of travelling wave solutions of the model are entirely determined by the basic reproduction ratio <inline-formula><tex-math id="M1">\begin{document}$ \mathcal{R}_{0} $\end{document}</tex-math></inline-formula>. As an application, we prove that if the localized amount of infectious individuals are introduced at the beginning, then the solution of such a system has an asymptotic spreading speed in large time and that is exactly coincident with the minimal wave speed.

Chemistry and complexity for solitons in AdS 5

Minimal D = 5 supergravity admits asymptotically globally AdS5 gravitational solitons (stationary, geodesically complete, horizonless spacetimes with positive mass). We show that, like asymptotically flat gravitational solitons, these solutions satisfy mass and mass variation formulas analogous to those satisfied by AdS black holes. A thermodynamic volume associated to the non-trivial topology of the spacetime plays an important role in this construction. We then consider these solitons within the holographic ‘complexity equals action’ and ‘complexity equals volume’ conjectures as simple examples of spacetimes with nontrivial rotation and topology. We find distinct behaviours for the volume and action, with the counterterm for null boundaries playing a significant role in the latter case. For large solitons we find that both proposals yield a complexity of formation proportional to a power of the thermodynamic volume, V 3/4. In fact, up to numerical prefactors, the result coincides with the analogous one for large black holes.

The Lp Minkowski problem for q-capacity

In the present paper, we first introduce the concepts of the Lpq-capacity measure and Lp mixed q-capacity and then prove some geometric properties of Lpq-capacity measure and a Lp Minkowski inequality for the q-capacity for any fixed p ⩾ 1 and q &gt; n. As an application of the Lp Minkowski inequality mentioned above, we establish a Hadamard variational formula for the q-capacity under p-sum for any fixed p ⩾ 1 and q &gt; n, which extends results of Akman et al. (Adv. Calc. Var. (in press)). With the Hadamard variational formula, variational method and Lp Minkowski inequality mentioned above, we prove the existence and uniqueness of the solution for the Lp Minkowski problem for the q-capacity which extends some beautiful results of Jerison (1996, Acta Math.176, 1–47), Colesanti et al. (2015, Adv. Math.285, 1511–588), Akman et al. (Mem. Amer. Math. Soc. (in press)) and Akman et al. (Adv. Calc. Var. (in press)). It is worth mentioning that our proof of Hadamard variational formula is based on Lp Minkowski inequality rather than the direct argument which was adopted by Akman (Adv. Calc. Var. (in press)). Moreover, as a consequence of Lp Minkowski inequality for q-capacity, we get an interesting isoperimetric inequality for q-capacity.

Diurnal Black-Sky Surface Albedo Parameterization of Snow

AbstractSurface albedo, the fraction of incoming solar radiation reflected hemispherically by the surface, is an essential climate variable (ECV) directly related to the energy budget of Earth. The presence and properties of snow cover alter surface albedo significantly, with variability in temporal scales reaching from seasonal to diurnal. The diurnal variation of snow albedo is typically parameterized with the solar zenith angle, but it cannot take into account asymmetry with respect to midday. Using the solar azimuth angle instead is suggested, since especially in the melting season the snow albedo varies highly asymmetrically during the day. To derive a general time- and latitude-independent formula, the azimuth angle values are normalized. Baseline Surface Radiation Network data are used to derive an empirical formula for the diurnal variation of snow black-sky surface albedo. The overall accuracy is on the order of 0.02, and the relative accuracy is about 3%.

Identification of the Polaron measure in strong coupling and the Pekar variational formula

The path measure corresponding to the Fröhlich polaron appearing in quantum statistical mechanics is defined as the tilted measure \begin{equation*}\mathrm{d}\widehat{\mathbb{P}}_{\varepsilon ,T}=\frac{1}{Z(\varepsilon ,T)}\exp \biggl\{\frac{1}{2}\int _{-T}^{T}\int _{-T}^{T}\frac{\varepsilon \mathrm{e}^{-\varepsilon \vert t-s\vert }}{\vert \omega (t)-\omega (s)\vert }\,\mathrm{d}s\,\mathrm{d}t\biggr\}\,\mathrm{d}\mathbb{P}.\end{equation*} Here, $\varepsilon >0$ is a constant known as the Kac parameter or the inverse-coupling parameter, and $\mathbb{P}$ is the distribution of the increments of the three-dimensional Brownian motion. In (Comm. Pure Appl. Math. 73 (2020) 350–383) it was shown that, when $\varepsilon >0$ is sufficiently small or sufficiently large, the (thermodynamic) limit $\lim_{T\to \infty }\widehat{\mathbb{P}}_{\varepsilon ,T}={\widehat{\mathbb{P}}}_{\varepsilon }$ exists as a process with stationary increments, and this limit was identified explicitly as a mixture of Gaussian processes. In the present article the strong coupling limit or the vanishing Kac parameter limit $\lim_{\varepsilon \to 0}\widehat{\mathbb{P}}_{\varepsilon }$ is investigated. It is shown that this limit exists and coincides with the increments of the so-called Pekar process, a stationary diffusion with generator $\frac{1}{2}\Delta +(\nabla \psi /\psi )\cdot \nabla $, where $\psi $ is the unique (up to spatial translations) maximizer of the Pekar variational problem \begin{equation*}g_{0}=\mathop{\mathrm{sup}}_{\Vert \psi \Vert _{2}=1}\biggl\{\int _{\mathbb{R}^{3}}\int _{\mathbb{R}^{3}}\psi ^{2}(x)\psi ^{2}(y)\vert x-y\vert ^{-1}\,\mathrm{d}x\,\mathrm{d}y-\frac{1}{2}\Vert \nabla \psi \Vert _{2}^{2}\biggr\}.\end{equation*} As the Pekar process was also earlier shown (Ann. Probab. 44 (2016) 3934–3964; Ann. Inst. Henri Poincaré Probab. Stat. 53 (2017) 2214–2228; Comm. Pure Appl. Math. 70 (2017) 1598–1629) to be the limiting object of the mean-field polaron measures, the present identification of the strong coupling limit is a rigorous justification of the mean-field approximation of the polaron problem (on the level of path measures) conjectured by Spohn in (Ann. Physics 175 (1987) 278–318). Replacing the Coulomb potential by continuous function vanishing at infinity and assuming uniqueness (modulo translations) of the relevant variational problem, our proof also shows that path measures coming from a Kac interaction of the above form with translation invariance in space converge to the increments of the corresponding mean-field model.

Projective synchronization of fuzzy memristive neural networks with pinning impulsive control

This paper is concerned with the problems of projective synchronization for fuzzy memristive neural networks (FMNNs) with pinning impulsive control. At first, the mathematical model of FMNNs is formulated on the basis of Takagi-Sugeno (T-S) fuzzy rules. According to the physical properties of memristor, the fact of parameters mismatch is also considered between the master and slave systems. Then, through introducing a novel type of pinning impulsive controllers and applying the Lyapunov-Krasovskii functional (LKF) method, some corresponding weak projective synchronization criteria are constructed. In order to handle the difficulties of error system discontinuous, the extended comparison principle based on the formula for variation of parameters and Cauchy matrix of linear impulsive system are introduced. Furthermore, complete projective synchronization criteria are also discussed under some given conditions. Finally, two simulation examples are given to show the effectiveness and validity of the obtained results.

Bending Analysis of Functionally Graded Nanoscale Plates by Using Nonlocal Mixed Variational Formula

This work is devoted to the bending analysis of functionally graded (FG) nano-scale plate by using the nonlocal mixed variational formula under simply supported edge conditions. According to Eringen’s nonlocal elasticity theory, the mixed formula is utilized in order to obtain the governing equations. The system of equations is derived by using the principle of virtual work. The governing equations include both the small and the mechanical effects. The impact of the small-scale parameter, aspect and thickness nano-scale plate ratios, and gradient index on the displacement and stresses are explored, numerically presented, and discussed in detail. Different comparisons are made to check the precision and validity of the bending outcomes obtained from the present analysis of FG nano-scale plates. Parametric examinations are then performed to inspect the impacts of the thickness of the plate on the by and large mechanical reaction of the practically evaluated plates. The displayed outcomes are valuable for the configuration procedures of keen structures and examination from materials.