Abstract:
Many important physical phenomena are described by wave or diffusion-wave type equations. Since these equations are linear, it would be useful to be able to use tools from the theory of linear signals and systems in solving related forward or inverse problems. In particular, the transform domain signal description from linear system theory has shown concrete promise for the solution of problems that are governed by a multidimensional wave field. The aim is to develop a unified framework for the description of wavefields via multidimensional signals. However, certain preliminary mathematical results are crucial for the development of this framework. This first paper on this topic thus introduces the mathematical foundations and proves some important mathematical results. The foundation of the framework starts with the inhomogeneous Helmholtz or pseudo-Helmholtz equation, which is the mathematical basis of a large class of wavefields. Application of the appropriate multi-dimensional Fourier transform leads to a transfer function description. To return to the physical spatial domain, certain mathematical results are necessary and these are presented and proved here as six fundamental theorems. These theorems are crucial for the evaluation of a certain class of improper integrals which arise in the evaluation of inverse multi-dimensional Fourier and Hankel transforms, upon which the framework is based. Subsequently, applications of these theorems are demonstrated, in particular for the derivation of Green's functions in different coordinate systems.

Abstract:
The use of linear relationships that can appear in heat transfer phenomena is described using a simple physicalexperimental situation in which the temperature evolution with time in a sample heated with low intensity continuouslight is measured. These questions should be included in the introductory physics curricula of science and engineeringstudies to teach aspects from different branches of physics (for example thermodynamics) and mathematics (rangingfrom functional analysis to differential equations).

Abstract:
We construct an adaptive asymptotically optimal in the classical norm of the space L(2) of square integrable functions non - parametrical multidimensional time defined signal regaining (adaptive filtration, noise canceller) on the background noise via multidimensional truncated Legendre expansion and optimal experience design. The two - dimensional case is known as a picture processing, picture analysis or image processing. We offer a two version of an confidence region building, also adaptive. Our estimates proposed by us have successfully passed experimental tests on problem by simulate of modeled with the use of pseudo-random numbers as well as on real data (of seismic signals etc.) for which our estimations of the different signals were compared with classical estimates obtained by the kernel or wavelets estimations method. The precision of proposed here estimations is better. Our adaptive truncation may be used also for the signal and image compression.

Abstract:
Circadian clocks gate cellular proliferation and, thereby, therapeutically target availability within proliferative pathways. This temporal coordination occurs within both cancerous and noncancerous proliferating tissues. The timing within the circadian cycle of the administration of drugs targeting proliferative pathways necessarily impacts the amount of damage done to proliferating tissues and cancers. Concurrently measuring target levels and associated key pathway components in normal and malignant tissues around the circadian clock provides a path toward a fuller understanding of the temporal relationships among the physiologic processes governing the therapeutic index of antiproliferative anticancer therapies. The temporal ordering among these relationships, paramount to determining causation, is less well understood using two- or three-dimensional representations. We have created multidimensional multimedia depictions of the temporal unfolding of putatively causative and the resultant therapeutic effects of a drug that specifically targets these ordered processes at specific times of the day. The systems and methods used to create these depictions are provided, as well as three example supplementary movies.

Abstract:
We represent a version of multidimensional quasilinear partial differential equation (PDE) together with large manifold of particular solutions given in an integral form. The dimensionality of constructed PDE can be arbitrary. We call it the $n$-wave type PDE, although the structure of its nonlinearity differs from that of the classical completely integrable (2+1)-dimensional $n$-wave equation. The richness of solution space to such a PDE is characterized by a set of arbitrary functions of several variables. However, this richness is not enough to provide the complete integrability, which is shown explicitly. We describe a class of multi-solitary wave solutions in details. Among examples of explicit particular solutions, we represent a lump-lattice solution depending on five independent variables. In Appendix, as an important supplemental material, we show that our nonlinear PDE is reducible from the more general multidimensional PDE which can be derived using the dressing method based on the linear integral equation with the kernel of a special type (a modification of the $\bar\partial$-problem). The dressing algorithm gives us a key for construction of higher order PDEs, although they are not discussed in this paper.

Abstract:
The aim of the present paper is to establish the multidimensional counterpart of the \textit{fourth moment criterion} for homogeneous sums in independent leptokurtic and mesokurtic random variables (that is, having positive and zero fourth cumulant, respectively), recently established in \cite{NPPS} in both the classical and in the free setting. As a consequence, the transfer principle for the Central limit Theorem between Wiener and Wigner chaos can be extended to a multidimensional transfer principle between vectors of homogeneous sums in independent commutative random variables with zero third moment and with non-negative fourth cumulant, and homogeneous sums in freely independent non-commutative random variables with non-negative fourth cumulant.

Abstract:
Due to the importance of boiling heat transfer in general, and boiling crisis in particular, for the analysis of operation and safety of both nuclear reactors and conventional thermal power systems, extensive efforts have been made in the past to develop a variety of methods and tools to evaluate the boiling heat transfer coefficient and to assess the onset of temperature excursion and critical heat flux (CHF) at various operating conditions of boiling channels. The objective of this paper is to present mathematical modeling concepts behind the development of mechanistic multidimensional models of low-quality forced convection boiling, including the mechanisms leading to temperature excursion and the onset of CHF.

Abstract:
In the theory of hyperbolic PDEs, the boundary-value problems with conditions on the entire boundary of the domain serve typically as the examples of the ill-posedness. The paper shows the unique solvability of the Dirichlet problem in the cylindric domain for the multidimensional wave equation. We also establish the criterion for the unique solvability of the equation.

Abstract:
In the framework of the Hartle-Hawking no-boundary proposal, we investigate quantum creation of the multidimensional universe with the cosmological constant $\Lambda$ but without matter fields. In this paper we solved the Wheeler-de Witt equation numerically. We find that the universe in which both of the spaces expand exponentially is the most probable in this model.

Abstract:
In this paper, we propose the Fourier frequency vector (FFV), inherently, associated with multidimensional Fourier transform. With the help of FFV, we are able to provide physical meaning of so called negative frequencies in multidimensional Fourier transform (MDFT), which in turn provide multidimensional spatial and space-time series analysis. The complex exponential representation of sinusoidal function always yields two frequencies, negative frequency corresponding to positive frequency and vice versa, in the multidimensional Fourier spectrum. Thus, using the MDFT, we propose multidimensional Hilbert transform (MDHT) and associated multidimensional analytic signal (MDAS) with following properties: (a) the extra and redundant positive, negative, or both frequencies, introduced due to complex exponential representation of multidimensional Fourier spectrum, are suppressed, (b) real part of MDAS is original signal, (c) real and imaginary part of MDAS are orthogonal, and (d) the magnitude envelope of a original signal is obtained as the magnitude of its associated MDAS, which is the instantaneous amplitude of the MDAS. The proposed MDHT and associated DMAS are generalization of the 1D HT and AS, respectively. We also provide the decomposition of an image into the AM-FM image model by the Fourier method and obtain explicit expression for the analytic image computation by 2DDFT.