Maximum Theorem Research Articles

Phase separation in quasi-incompressible fluids: Cahn–Hilliard model in the Cattaneo–Maxwell framework

In this paper, we propose a mathematical model of phase separation for a quasi-incompressible binary mixture where the spinodal decomposition is induced by an heat flux governed by the Cattaneo–Maxwell equation. As usual, the phase separation is considered in the framework of phase-field modeling so that the transition is described by an additional field, the mixture concentration c. The evolution of the mixture concentration is described by the Cahn–Hilliard equation, and in our model, it is coupled with the Navier–Stokes equation. Since thermal effect is included, the whole set of evolution equations is set up for the velocity, the mixture concentration, the temperature and the heat flux. The model is compatible with thermodynamics and a maximum theorem holds.

The Schwarz lemma in Clifford analysis

In this paper, we first give the Cauchy type integral representation for harmonic functions in the Clifford analysis framework, and by using integral representations for harmonic functions in Clifford analysis, the Poisson integral formula for harmonic functions is represented. As its application, the mean value theorems and the maximum modulus theorem for Clifford-valued harmonic functions are presented. Second, some properties of Möbius transformations are given, and a close relation between the monogenic functions and Möbius transformations is shown. Finally, by using the integral representations for harmonic functions and the properties of Möbius transformations, Schwarz type lemmas for harmonic functions and monogenic functions are established.

A maximum degree theorem for diameter-2-critical graphs

AbstractA graph is diameter-2-critical if its diameter is two and the deletion of any edge increases the diameter. Let G be a diameter-2-critical graph of order n. Murty and Simon conjectured that the number of edges in G is at most ⌊n 2/4⌋ and that the extremal graphs are the complete bipartite graphs K ⌊n/2⌋,⌊n/2⌉. Fan [Discrete Math. 67 (1987), 235–240] proved the conjecture for n ≤ 24 and for n = 26, while Füredi [J. Graph Theory 16 (1992), 81–98] proved the conjecture for n > n 0 where n 0 is a tower of 2’s of height about 1014. The conjecture has yet to be proven for other values of n. Let Δ denote the maximum degree of G. We prove the following maximum degree theorems for diameter-2-critical graphs. If Δ ≥ 0.7 n, then the Murty-Simon Conjecture is true. If n ≥ 2000 and Δ ≥ 0.6789 n, then the Murty-Simon Conjecture is true.

Bergeʼs maximum theorem for noncompact image sets

This note generalizes Bergeʼs maximum theorem to noncompact image sets. It also clarifies the results from Feinberg, Kasyanov and Zadoianchuk (2013) [7] on the extension to noncompact image sets of another Bergeʼs theorem, that states semi-continuity of value functions. Here we explain that the notion of a K-inf-compact function introduced there is applicable to metrizable topological spaces and to more general compactly generated topological spaces. For Hausdorff topological spaces we introduce the notion of a KN-inf-compact function (N stands for “nets” in K-inf-compactness), which coincides with K-inf-compactness for compactly generated and, in particular, for metrizable topological spaces.

Plant refuses driven biochar: Application as metal adsorbent from acidic solutions

Biochar prepared from aromatic spent was used as adsorbent for assessing its removal capacity of cadmium, chromium, copper and lead from aqueous acidic solutions. For the optimization of the processes, separate experiments were setup in fixed bed columns of biochar prepared from different biomasses in variable sizes at different temperatures, metal concentrations, flow rate and time. The effect of the above parameters on adsorption of metals was assessed in terms of maximum adsorption capacity, kinetics, theorem and thermodynamics. Results show that maximum removal of each metal was 60mg/g. The adsorption equilibrium data obeyed the Freundlich model and the kinetic data were well described by the pseudo-second-order model. The adsorption process is believed to proceed by an initial surface adsorption followed by intra-particle diffusion. In this regard to the proposed mechanism, modeling results implied that exchange of the hydrogen occurs during the low loading of metal. Opposite is true for the calcium, magnesium and sodium ions. Thermodynamic studies revealed the feasibility and endothermic nature of the system. Treatment of acidic mine water with biochar suggests that it buffers the acid and is capable of efficient removal of these metals.

Phase transition and separation in compressible Cahn-Hilliard fluids

The paper provides a scheme for phase separation and transition by accounting for diffusion, dynamic equations and consistency with thermodynamics. The constituents are compressible fluids thus improving the model of a previous approach. Moreover a possible saturation effect for the concentration of a constituent is made explicit. The mass densities of the constituents are independent of temperature. The evolution of concentration is described by the standard equation for mixtures but the balance of energy and entropy of the mixture are stated as for a single constituent. However, due to the non-simple character of the mixture, an extra-energy flux is allowed to occur. Also motion and diffusion effects are considered by letting the stress in the mixture have additive viscous terms and, remarkably, the chemical potential contains a quadratic term in the stretching tensor. As a result a whole set of evolution equations is set up for the concentration, the velocity, and the temperature. Shear-induced mixing and demixing are examined. A maximum theorem is proved which implies that the concentration of the mixture has values from 0 to 1 as is required from the physical standpoint.

Mode-coupling theory for multiple decay channels

We investigate the properties of a class of mode-coupling equations for the glass transition where the density mode decays into multiple relaxation channels. We prove the existence and uniqueness of the solutions for Newtonian as well as Brownian dynamics and demonstrate that they fulfil the requirements of correlation functions; in the latter case the solutions are purely relaxational. Furthermore, we construct an effective mode-coupling functional, which allows us to map the theory to the case of a single decay channel, such that the covariance principle found for the mode-coupling theory for simple liquids is properly generalized. This in turn allows establishment of the maximum theorem stating that long-time limits of mode-coupling solutions can be calculated as maximal solutions of a fixed-point equation without relying on the dynamic solutions.

A mapping-based tree similarity algorithm and its application to ontology alignment

A mapping-based tree similarity algorithm is proposed for matching concept trees in ontology alignment to integrate various information sources in the Semantic Web. Concepts regarding classes and properties are the most critical ontological elements and metadata. First, the similarity between the individual concepts of each type is defined. These concept systems, which are considered as the foundation of ontology, are described as tree modes for overall comparison. Based on the minimal cost of edit operations, previous tree similarity measuring approaches are extremely complicated because three or four edit operations are involved. Moreover, such approaches ignore the similarity among single nodes. In the proposed algorithm, node similarity, instead of changing operation, is adopted and the inserting and deleting operation is omitted. The proposed algorithm is more concise and effective because it satisfies the maximum mapping theorem without damaging tree isomorphism. The algorithm is resolved and realized by a dynamic programming scheme. Then, the algorithm is independently used to compare class and property trees, and their mapping concept sets are regarded as the main part of the ontology alignment. Demonstration examples are used to prove the effectiveness and feasibility of the algorithm in ontology alignment.

On the Stokes problem in exterior domains: The maximum modulus theorem

We study the Stokes initial boundary value problem, in$(0,T) \times Ω$, where $Ω \subseteq \mathbb{R}^n$, $n\geq3$, is an exteriordomain, assuming that the initial data belongs to $L^\infty(Ω)$and has null divergence in weak sense. We prove the maximum modulustheorem for the corresponding solutions. Crucial for the proof ofthis result is the analogous one proved by Abe-Giga for boundeddomains. Our proof is developed by duality arguments and employingthe semigroup properties of the resolving operator defined on$L^1(Ω)$. Our results are similar to the ones proved by Solonnikovby means of the potential theory.

An Extention of the Hadamard-Type Inequality for a Convex Function Defined on Modulus of Complex Integral Functions

In this paper we extend the Hadamard's type inequalities for convex functions defined on the modulus of integral functions in complex field. Firstly, by using the Principal of maximum modulus theorem we show that $M(r)$ and $lnM(r)$ are continuous and convex functions for any non-negative values of $r$. Finally we derive two inequalities analogous to well known Hadamard's inequality by using elementary analysis.

Limit theorems for random maximum of independent and non-identically distributed random vectors

In this paper, we study the weak convergence of the random maximum of independent and non-identical random vectors. When the random sample size is assumed to be independent of the basic variables and its distribution function is assumed to converge weakly to a non-degenerate limit, the necessary and sufficient conditions for the weak convergence of the random maximum are derived. An illustrative example is given.

A phase field model for brine channels in sea ice

In this paper, we present a phenomenological mathematical model for describing the features of the brine channels in sea ice. The differential system is composed of the Ginzburg–Landau and Cahn–Hilliard equations, in addition to the heat equation, that controls the ice–liquid phase transition by the temperature and hence the establishment of brine channels. The compatibility of this system with the thermodynamic laws and a maximum theorem is proved.

Fluid pressure control on splay fault activation in accretionary prism based on the maximum strength theorem with application to the Nankai wedge

The maximum strength theorem is applied to determine the pressure conditions in accretionary wedges responsible for the activation of splay faults (SF). This theorem, based on the limit analysis developed in soil mechanics, selects among several modes of deformation the one prevailing for a given pressure distribution. It applies for frictional rock properties and for an arbitrary wedge geometry. In particular, the décollement is assumed here to be partitioned into two mechanically distinct regions, the internal region being part of the seismogenic zone. The considered modes of deformation include the activation of a thrust fold rooting on the décollement and outcropping anywhere on the topography. This mode, if close to the transition between the seismic and the aseismic region of the décollement is in competition with the SF which constitutes also a potential mode of deformation. The dominance of the various modes of deformation is presented in stability maps in the plane spanned by the pressure ratios of the two regions of the décollement (pore pressure divided by lithostatic pressure, both referenced to the sea floor). Several maps are proposed for the particular case of the Kumano transect of Nankai wedge, South-West Japan, varying the SF pressure ratio, the position of the SF root on the décollement with respect to the transition between the seismic and aseismic regions as well as the respective length of these two regions. In addition, these stability maps are suggesting various scenarios to explain the changes through time in the prevailing mode of deformation in an accretionary wedges. For the Kumano transect, three scenarios could explain the observed deactivation of the SF, one of them being the seaward migration of the décollement transition point.

Physical collisions and the maximum power theorem: an analogy between mechanical and electrical situations

The maximum power theorem is a useful extension to work on EMF and internal resistance at school level. Furthermore, a very simple physical collision model can be used to show equivalent mathematical patterns to those found with the maximum power theorem and to emphasize fundamental links to ideas of impedance matching.

Coherent risk measures in general economic models and price bubbles

In this article we study coherent risk measures in general economic models where the set of financial positions is an ordered Banach space E and the safe asset an order unit x0 of E. First we study some properties of risk measures. We show that the set of normalized (with respect to x0) price systems is weak star compact and by using this result we prove a maximum attainment representation theorem which improves the one of Jaschke and Küchler (2001). Also we study how a risk measure changes under different safe assets and we show a kind of equivalence between these risk measures. In the sequel we study subspaces of E consisting of financial positions of risk greater or equal to zero and we call these subspaces unsure. We find some criteria and we give examples of these subspaces. In the last section, we combine the unsure subspaces with the theory of price-bubbles of Gilles and LeRoy (1992).In this study we use the theory of cones (ordered spaces). This theory allows us to generalize basic results and provides new proofs and ideas in the theory of risk measures.

Phase Transition In Liquid 4He by a Mean Field Model

In this work the transition of 4He at the lambda line in presence of a Cattaneo-Maxwell heat flux is studied. A hydrodynamical model is formulated, which chooses as fundamental fields the velocity, the temperature, the heat flux and a phase field function f, for which a time dependent Ginzburg-Landau equation is proposed. Using this model we are able to describe the phase transition and to obtain the pressure-temperature phase diagram which represents the transition, the thermodynamic restrictions and a maximum theorem for the phase field.

Improper Gaussian Signaling on the Two-User SISO Interference Channel

On a single-input-single-out (SISO) interference channel (IC), conventional non-cooperative strategies encourage players selfishly maximizing their transmit data rates, neglecting the deficit of performance caused by and to other players. In the case of proper complex Gaussian noise, the maximum entropy theorem shows that the best-response strategy is to transmit with proper signals (symmetric complex Gaussian symbols). However, such equilibrium leads to degrees-of-freedom zero due to the saturation of interference. With improper signals (asymmetric complex Gaussian symbols), an extra freedom of optimization is available. In this paper, we study the impact of improper signaling on the 2-user SISO IC. We explore the achievable rate region with non-cooperative strategies by computing a Nash equilibrium of a non-cooperative game with improper signaling. Then, assuming cooperation between players, we study the achievable rate region of improper signals. We propose the usage of improper rank one signals for their simplicity and ease of implementation. Despite their simplicity, rank one signals achieve close to optimal sum rate compared to full rank improper signals. We characterize the Pareto boundary, the outer-boundary of the achievable rate region, of improper rank one signals with a single real-valued parameter; we compute the closed-form solution of the Pareto boundary with the non-zero-forcing strategies, the maximum sum rate point and the max-min fairness solution with zero-forcing strategies. Analysis on the extreme SNR regimes shows that proper signals maximize the wide-band slope of spectral efficiency whereas improper signals optimize the high-SNR power offset.

On the Maximum Theorem: A Constructive Analysis

We examine the maximum theorem by Berge from the point of view of Bishop style constructive mathematics. We will show an approximate version of the maximum theorem and the maximum theorem for functions with sequentially locally at most one maximum. Keywords—Maximum theorem, Constructive mathematics, Sequentially locally at most one maximum.

A constructive version of Ky Fan's coincidence theorem

In this paper we examine Ky Fan's coincidence theorem from the point of view of Bishop style constructive mathematics using constructive versions of Kakutani's fixed point theorem, Berge's maximum theorem and the separating hyperplane theorem.

Complex-Valued Random Vectors and Channels: Entropy, Divergence, and Capacity

Recent research has demonstrated significant achievable performance gains by exploiting circularity/non-circularity or propeness/improperness of complex-valued signals. In this paper, we investigate the influence of these properties on important information theoretic quantities such as entropy, divergence, and capacity. We prove two maximum entropy theorems that strengthen previously known results. The proof of the former theorem is based on the so-called circular analog of a given complex-valued random vector. Its introduction is supported by a characterization theorem that employs a minimum Kullback-Leibler divergence criterion. In the proof of latter theorem, on the other hand, results about the second-order structure of complex-valued random vectors are exploited. Furthermore, we address the capacity of multiple-input multiple-output (MIMO) channels. Regardless of the specific distribution of the channel parameters (noise vector and channel matrix, if modeled as random), we show that the capacity-achieving input vector is circular for a broad range of MIMO channels (including coherent and noncoherent scenarios). Finally, we investigate the situation of an improper and Gaussian distributed noise vector. We compute both capacity and capacity-achieving input vector and show that improperness increases capacity, provided that the complementary covariance matrix is exploited. Otherwise, a capacity loss occurs, for which we derive an explicit expression.