We present a new nonlinear version of the well-known Black-Scholes model for option pricing in financial mathematics. The nonlinear Black-Scholes partial differential equation is based on the ...quasilinear diffusion term with the p-Laplace operator \(\Delta_p\) for \(1 < p < \infty\). The existence and uniqueness of a weak solution in a weighted Sobolev space is proved, first, by methods for nonlinear parabolic problems using the Gel'fand triplet and, alternatively, by a method based on nonlinear semigroups. Finally, possible choices of other weighted Sobolev spaces are discussed to produce a function space setting more realistic in financial mathematics.
See also https://ejde.math.txstate.edu/special/02/t1/abstr.html
We study analytic smooth solutions of a general, strongly parabolic semilinear Cauchy problem of 2m-th order in \(\mathbb{R}^N\times (0,T)\) with analytic coefficients (in space and time variables) ...and analytic initial data (in space variables).They are expressed in terms of holomorphic continuation of global (weak) solutions to the system valued in a suitable Besov interpolation space of \(B^{s;p,p}\)-type at every time moment \(t\in 0,T\). Given \(0 < T'< T\leq \infty\), it is proved that any \(B^{s;p,p}\)-type solution \(u: \mathbb{R}^N\times (0,T)\to \mathbb{C}^M\) with analytic initial data possesses a bounded holomorphic continuation \(u(x + \mathrm{i}y, \sigma + \mathrm{i}\tau)\) into a complex domain in \(\mathbb{C}^N\times \mathbb{C}\) defined by \((x,\sigma)\in \mathbb{R}^N\times (T',T)\), \(|y| < A'\) and \(|\tau | < B'\), where \(A', B'> 0\) are constants depending upon~\(T'\). The proof uses the extension of a weak solution to a \(B^{s;p,p}\)-type solution in a domain in\(\mathbb{C}^N\times \mathbb{C}\), such that this extension satisfies the Cauchy-Riemann equations. The holomorphic extension is obtained with a help from holomorphic semigroups and maximal regularity theory for parabolic problems in Besov interpolation spaces of \(B^{s;p,p}\)-type imbedded (densely and continuously) into an \(L^p\)-type Lebesgue space. Applications include \emph{risk models} for European options in Mathematical Finance.
For more information see https://ejde.math.txstate.edu/special/01/b1/abstr.html
Lookism is a term used to describe discrimination based on the physical appearance of a person. We suppose that the social impact of lookism is a philosophical issue, because, from this perspective, ...attractive people have an advantage over others. The first line of our argumentation involves the issue of lookism as a global ethical and aesthetical phenomenon. A person’s attractiveness has a significant impact on the social and public status of this individual. The common view in society is that it is good to be more attractive and healthier. This concept generates several ethical questions about human aesthetical identity, health, authenticity, and integrity in society. It seems that this unequal treatment causes discrimination, diminishes self-confidence, and lowers the chance of a job or social enforcement for many human beings. Currently, aesthetic improvements are being made through plastic surgery. There is no place on the human body that we cannot improve with plastic surgery or aesthetic medicine. We should not forget that it may result in the problem of elitism, in dividing people into primary and secondary categories. The second line of our argumentation involves a particular case of lookism: Melanie Gaydos. A woman that is considered to be a model with a unique look.
The weak and strong comparison principles (WCP and SCP, respectively) are investigated for quasilinear elliptic boundary value problems with the p-Laplacian in one space dimension, ...Δp(u)=defddx(|u′|p−2u′). We treat the “degenerate” case of 2<p<∞ and allow also for the nontrivial convection velocityb:−1,1→R in the underlying domain Ω=(−1,1). We establish the WCP under a rather general, “natural sufficient condition” on the convection velocity, b(x), and the reaction function, φ(x,u). Furthermore, we establish also the SCP under a number of various additional hypotheses. In contrast, with these hypotheses being violated, we construct also a few rather natural counterexamples to the SCP and discuss their applications to an interesting classical problem of fluid flow in porous medium, “seepage flow of fluids in inclined bed”. Our methods are based on a mixture of classical and new techniques.
We consider a one-dimensional population genetics model for the advance of an advantageous gene. The model is described by the semilinear Fisher equation with unbalanced bistable
non-Lipschitzian
...nonlinearity
f
(
u
). The “nonsmoothness” of
f
allows for the appearance of travelling waves with a new, more realistic profile. We study existence, uniqueness, and long-time asymptotic behavior of the solutions
u
(
x
,
t
),
(
x
,
t
)
∈
R
×
R
+
. We prove also the existence and uniqueness (up to a spatial shift) of a travelling wave
U
. Our main result is the uniform convergence (for
x
∈
R
) of every solution
u
(
x
,
t
) of the Cauchy problem to a single travelling wave
U
(
x
-
c
t
+
ζ
)
as
t
→
∞
. The speed
c
and the travelling wave
U
are determined uniquely by
f
, whereas the shift
ζ
is determined by the initial data.
We consider a one-dimensional reaction–diffusion equation of Fisher–Kolmogoroff–Petrovsky–Piscounoff type. We investigate the effect of the interaction between the nonlinear diffusion coefficient and ...the reaction term on the existence and non-existence of travelling waves. Our diffusion coefficient is allowed to be degenerate or singular at both equilibrium points, 0 and 1, while the reaction term need not be differentiable. These facts influence the existence and qualitative properties of travelling waves in a substantial way.
Approximately 20% of sleeping sickness patients exhibit respiratory complications, however, with a largely unknown role of the parasite. Here we show that tsetse fly-transmitted Trypanosoma brucei ...parasites rapidly and permanently colonize the lungs and occupy the extravascular spaces surrounding the blood vessels of the alveoli and bronchi. They are present as nests of multiplying parasites exhibiting close interactions with collagen and active secretion of extracellular vesicles. The local immune response shows a substantial increase of monocytes, macrophages, dendritic cells and γδ and activated αβ T cells and a later influx of neutrophils. Interestingly, parasite presence results in a significant reduction of B cells, eosinophils and natural killer cells. T. brucei infected mice show no infection-associated pulmonary dysfunction, mirroring the limited pulmonary clinical complications during sleeping sickness. However, the substantial reduction of the various immune cells may render individuals more susceptible to opportunistic infections, as evident by a co-infection experiment with respiratory syncytial virus. Collectively, these observations provide insights into a largely overlooked target organ, and may trigger new diagnostic and supportive therapeutic approaches for sleeping sickness.
Origin of the p-Laplacian and A. Missbach Jiri Benedikt; Petr Girg; Lukas Kotrla ...
Electronic journal of differential equations,
01/2018, Volume:
2018, Issue:
16
Journal Article
Peer reviewed
Open access
We describe the historical process of derivation of the p-Laplace operator from a nonlinear Darcy law and the continuity equation. The story begins with nonlinear flows in channels and ditches. As ...the nonlinear Darcy law we use the power law discovered by Smreker and verified in experiments by Missbach for flows through porous media in one space dimension. These results were generalized by Christianovitch and Leibenson to porous media in higher space dimensions. We provide a brief description of Missbach's experiments.
Wolbachia is a genus of endosymbiotic α-Proteobacteria infecting a wide range of arthropods and filarial nematodes. Wolbachia is able to induce reproductive abnormalities such as cytoplasmic ...incompatibility (CI), thelytokous parthenogenesis, feminization and male killing, thus affecting biology, ecology and evolution of its hosts. The bacterial group has prompted research regarding its potential for the control of agricultural and medical disease vectors, including Glossina spp., which transmits African trypanosomes, the causative agents of sleeping sickness in humans and nagana in animals.
In the present study, we employed a Wolbachia specific 16S rRNA PCR assay to investigate the presence of Wolbachia in six different laboratory stocks as well as in natural populations of nine different Glossina species originating from 10 African countries. Wolbachia was prevalent in Glossina morsitans morsitans, G. morsitans centralis and G. austeni populations. It was also detected in G. brevipalpis, and, for the first time, in G. pallidipes and G. palpalis gambiensis. On the other hand, Wolbachia was not found in G. p. palpalis, G. fuscipes fuscipes and G. tachinoides. Wolbachia infections of different laboratory and natural populations of Glossina species were characterized using 16S rRNA, the wsp (Wolbachia Surface Protein) gene and MLST (Multi Locus Sequence Typing) gene markers. This analysis led to the detection of horizontal gene transfer events, in which Wobachia genes were inserted into the tsetse flies fly nuclear genome.
Wolbachia infections were detected in both laboratory and natural populations of several different Glossina species. The characterization of these Wolbachia strains promises to lead to a deeper insight in tsetse flies-Wolbachia interactions, which is essential for the development and use of Wolbachia-based biological control methods.
Analytic smoothing properties of a general, strongly parabolic linear Cauchy problem of second order in RN×(0,T) with analytic coefficients (in space and time variables) are investigated. They are ...expressed in terms of holomorphic continuation of global (weak) L2-type solutions to the system. Given 0<T′<T⩽∞, it is proved that any L2-type solution u:RN×(0,T)→RM possesses a bounded holomorphic continuation u(x+iy,σ+iτ) into a complex domain in CN×C defined by (x,σ)∈RN×(T′,T), |y|<A′ and |τ|<B′, where A′,B′>0 are constants depending upon T′. The proof uses the extension of a solution to an L2-type solution in a domain in CN×C, such that this extension satisfies the Cauchy–Riemann equations. The holomorphic extension is thus obtained in a Hardy space H2. Applications include market completion by European options in Finance.
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