spectra of digraphs in this primarily expository paper we survey classical and some more recent results on the spectra of digraphs equivalently the spectra of 0 1 matrices with emphasis on the spectral radius

excursions for polymers in elongational flows the study of the configurational statistical mechanics of polymers in elongational flows requires the evaluation of quadratic path integrals here we present a new technique for the evaluation of such path integrals the method relies on ito excursion theory and the ray knight theorem from probability theory in addition to providing a powerful computational method in standard cases the method generalizes to deal with quadratic path integrals for branching processes

an hp adaptive mixed discontinuous galerkin fem for nearly incompressible linear elasticity we develop the a posteriori error estimation of mixed hp version discontinuous galerkin finite element methods for nearly incompressible elasticity problems in two space dimensions computable upper and lower bounds on the error measured in terms of a natural mesh dependent energy norm are derived the bounds are explicit in the local mesh sizes and approximation orders and are independent of the locking parameter a series of numerical experiments are presented which demonstrate the performance of the proposed error estimator within an automatic hp adaptive refinement procedure

local regularization for the nonlinear inverse autoconvolution problem we develop a local regularization theory for the nonlinear inverse autoconvolution problem unlike classical regularization techniques such as tikhonov regularization this theory provides regularization methods that preserve the causal nature of the autoconvolution problem allowing for fast sequential numerical solution cal o rn 2 r 2n flops where r ll n for the method discussed in this paper as applied to the nonlinear problem in comparison the cost for tikhonov regularization applied to a general linear problem is cal o n 3 flops we prove the convergence of the regularized solutions to the true solution as the noise level in the data shrinks to zero and supply convergence rates for the cases of both l 2 and continuous data we propose several regularization methods and provide a theoretical basis for their convergence of note is that this class of methods does not require an initial guess of the unknown solution our numerical results confirm the effectiveness of the methods with results comparing favorably to numerical examples found in the literature for the autoconvolution problem e g g fleischer r gorenflo and b hofmann zamm z angew math mech 79 1999 pp 149 159 for examples using tikhonov regularization with total variation constraints and j janno inverse problems 16 2000 pp 333 348 for examples using the method of lavrent ev this especially seems to be true when it comes to the recovery of sharp features in the unknown solution we also show the effectiveness of our method in cases not covered by the theory

semiflexible polymers in straining flows we introduce a model for semiflexible polymer chains based on the integral of an appropriate gaussian process the stiffness is characterized physically by adding a bending energy the degree of stiffness in the polymer chain is quantified by means of a parameter and as this parameter tends to infinity the limiting case reduces to the brownian model of completely flexible chains studied in earlier work the calculation of the partition function for the configuration statistical mechanics i e the distribution of shapes of such polymers in elongational flow or quadratic potentials is equivalent to the probabilistic problem of finding the law of a quadratic functional of the associated gaussian process an exact formula for the partition function is presented however in practice this formula is too complicated for most computations we therefore develop an asymptotic expansion for the partition function in terms of the stiffness parameter and obtain the first order term which gives the first order deviation from the completely flexible case in addition to the partition function the method presented here can also deal with other quadratic functionals such as the stochastic area associated with two polymer chains

brownian excursions on combs in this paper we use techniques from ito excursion theory to analyze brownian motion on generalized combs ito excursion theory is a little known area of probability theory and we therefore present a brief introduction for the uninitiated a general method for analyzing transport along the backbone of the comb is demonstrated and the specific case of a comb whose teeth are scaling branching trees is examined we then present a recursive method for evaluating the distribution of the first passage times on hierarchical combs

on the distribution of estimators of diffusion constants for brownian motion we discuss the distribution of various estimators for extracting the diffusion constant of single brownian trajectories obtained by fitting the squared displacement of the trajectory the analysis of the problem can be framed in terms of quadratic functionals of brownian motion that correspond to the euclidean path integral for simple harmonic oscillators with time dependent frequencies explicit analytical results are given for the distribution of the diffusion constant estimator in a number of cases and our results are confirmed by numerical simulations

an augmented lagrangian method for identifying discontinuous parameters in elliptic systems the identification of discontinuous parameters in elliptic systems is formulated as a constrained minimization problem combining the output least squares and the equation error method the minimization problem is then proved to be equivalent to the saddle point problem of an augmented lagrangian the finite element method is used to discretize the saddle point problem and the convergence of the discretization is also proved finally an uzawa algorithm is suggested for solving the discrete saddle point problem and is shown to be globally convergent

the skew energy of a digraph we are interested in the energy of the skew adjacency matrix of a directed graph d which is simply called the skew energy of d in this paper properties of the skew energy of d are studied in particular a sharp upper bound for the skew energy of d is derived in terms of the order of d and the maximum degree of its underlying undirected graph an infinite family of digraphs attaining the maximum skew energy is constructed moreover the skew energy of a directed tree is independent of its orientation and interestingly it is equal to the energy of the underlying undirected tree skew energies of directed cycles under different orientations are also computed some open problems are presented

optimal estimates of the diffusion coefficient of a single brownian trajectory modern developments in microscopy and image processing are revolutionizing areas of physics chemistry and biology as nanoscale objects can be tracked with unprecedented accuracy the goal of single particle tracking is to determine the interaction between the particle and its environment the price paid for having a direct visualization of a single particle is a consequent lack of statistics here we address the optimal way of extracting diffusion constants from single trajectories for pure brownian motion it is shown that the maximum likelihood estimator is much more efficient than the commonly used least squares estimate furthermore we investigate the effect of disorder on the distribution of estimated diffusion constants and show that it increases the probability of observing estimates much smaller than the true average value

ordering of oriented unicyclic graphs by skew energies ordering of the oriented unicyclic graphs with n vertices according to their minimal skew energies is investigated we derive the first 22 oriented unicyclic graphs for n 1784 and a series of oriented unicyclic graphs for 3 n 1783

on an inverse problem from magnetic resonance elastic imaging the imaging problem of elastography is an inverse problem the nature of an inverse problem is that it is ill conditioned we consider properties of the mathematical map which describes how the elastic properties of the tissue being reconstructed vary with the field measured by magnetic resonance imaging mri this map is a nonlinear mapping and our interest is in proving certain conditioning and regularity results for this operator which occurs implicitly in this problem of imaging in elastography in this treatment we consider the tissue to be linearly elastic isotropic and spatially heterogeneous we determine the conditioning of this problem of function reconstruction in particular for the stiffness function we further examine the conditioning when determining both stiffness and density we examine the frechet derivative of the nonlinear mapping which enables us to describe the properties of how the field affects the individual maps to the stiffness and density functions we illustrate how use of the implicit function theorem can considerably simplify the analysis of frechet differentiability and regularity properties of this underlying operator we present new results which show that the stiffness map is mildly ill posed whereas the density map suffers from medium ill conditioning computational work has been done previously to study the sensitivity of these maps but our work here is analytical the validity of the newton kantorovich and optimization methods for the computational solution of this inverse problem is directly linked to the frechet differentiability of the appropriate nonlinear operator which we justify

5 regular oriented graphs with optimum skew energy abstract let g be a simple undirected graph and g be the corresponding oriented graph of g with the orientation the skew energy of g denoted by e s g is defined as the sum of the singular values of the skew adjacency matrix s g in 2010 adiga et al certified that s g n where is the maximum degree of g of order n it has been shown that every 5 regular oriented graph with optimum skew energy has even neighborhood property that is each pair of neighborhoods of a graph have even number of common vertices in this paper we characterize all connected 5 regular graphs of order n with this property moreover we determine all connected 5 regular oriented graphs of order n with maximum skew energy

symmetry breaking between statistically equivalent independent channels in few channel chaotic scattering we study the distribution function p ensuremath omega of the random variable ensuremath omega ensuremath tau 1 ensuremath tau 1 ensuremath cdots ensuremath tau n where ensuremath tau k s are the partial wigner delay times for chaotic scattering in a disordered system with n independent statistically equivalent channels in this case ensuremath tau k s are independent and identically distributed random variables with a distribution ensuremath psi ensuremath tau characterized by a fat power law intermediate tail ensuremath sim 1 ensuremath tau 1 ensuremath mu truncated by an exponential or a log normal function of ensuremath tau for n 2 and n 3 we observe a surprisingly rich behavior of p ensuremath omega revealing a breakdown of the symmetry between identical independent channels for n 2 numerical simulations of the quasi one dimensional anderson model confirm our findings

skew rank of an oriented graph in terms of matching number abstract an oriented graph g is a digraph without loops and multiple arcs where g is called the underlying graph of g let s g denote the skew adjacency matrix of g the rank of s g is called the skew rank of g denoted by sr g which is even since s g is skew symmetric li and yu 2015 12 proved that the skew rank of an oriented unicyclic graph g is either 2 m g 2 or 2 m g where m g denotes the matching number of g in this paper we extend this result to general cases it is proved that the skew rank of an oriented connected graph g is an even integer satisfying 2 m g 2 g sr g 2 m g where g e g v g 1 is the number of fundamental cycles also called the first betti number besides the oriented graphs satisfying sr g 2 m g 2 g are characterized definitely

constitutive error based material parameter estimation procedure for hyperelastic material abstract we have explored hyperelastic material parameter identification problem via minimizing trace of the error in constitutive relation function from partial and corrupted displacement measurements under quasi static loading the inverse problem is posed as an optimization problem where the cost function measures the error in constitutive relation due to a constitutively incompatible pair of stress and strain field incompatibility of stress and strain arises due to dissimilar constraints is imposed in generating these fields an alternating approach is adopted to solve the optimization problem a simple yet effective penalty based approach is adopted to generate strain field that weakly satisfy available measurements while the regularization due to illposedness is achieved through the penalization parameter this also allows us to consider partial and corrupted data map in our reconstruction strategy after generating incompatible fields desired reconstruction is obtained via minimizing the constitutive disparity in the process we obtain explicit parameter update formula finally we show the applicability and computational capabilities of the proposed method via limited number of numerical experiments

on upper bounds for the energy of digraphs abstract the energy of a digraph d is defined as e d i 1 n re z i where z 1 z 2 z n are the possibly complex eigenvalues of d in this paper we first give an improved lower bound on the spectral radius of the digraph d using this result we obtain a new upper bound on the energy e d and characterize some extreme digraphs which attain this upper bound this result theoretically improves and generalizes some known results

some brownian functionals and their laws we develop some topics about brownian motion with a particular emphasis on the study of principal values of brownian local times we show some links between principal values and doob s h transforms of brownian motion for nonpositive harmonic functions h we also give a survey and complement some martingale approaches to ray knight theorems for local times

a new convex inversion framework for parameter identification in saddle point problems with an application to the elasticity imaging inverse problem of predicting tumor location this work presents a thorough theoretical and numerical analysis of the elasticity imaging inverse problem of tumor identification in the soft tissue of the human body beyond the obvious merits of its applications this problem also presents significant mathematical challenges the near incompressibility inherent in the model of linear elasticity in the body gives rise to the locking effect and necessitates a unique treatment of both the direct and inverse problems a general optimization framework for the identification of parameters in saddle point problems is presented along with a new modified output least squares mols objective functional the mols functional is shown to be convex thus overcoming the nonconvexity of the classical output least squares ols functional and the new framework is shown to be capable of accommodating both smooth and discontinuous parameters generalized derivative formulas for the coefficient to solution map are also given along with a complete convergence analysis

oriented unicyclic graphs with the first n 92 largest skew energies abstract let g be an oriented graph obtained by assigning an orientation to the edge set of a simple undirected graph g such that g becomes a directed graph let s g be the skew adjacency matrix of g the skew energy of g is defined as the sum of the absolute values of all eigenvalues of s g in this paper we provide a new method to compare the skew energies of two oriented graphs whose skew characteristic polynomials satisfy a given recurrence relation and determine the oriented unicyclic graphs of order n with the first n 9 2 largest skew energies for n 31

least squares methods for linear elasticity this paper develops least squares methods for the solution of linear elastic problems in both two and three dimensions our main approach is defined by simply applying the l2 norm least squares principle to a stress displacement system the constitutive and the equilibrium equations it is shown that the homogeneous least squares functional is elliptic and continuous in the h rm div omega d times h 1 omega d norm this immediately implies optimal error estimates for finite element subspaces of h rm div omega d times h 1 omega d it admits optimal multigrid solution methods as well if raviart thomas finite element spaces are used to approximate the stress tensor our method does not degrade when the material properties approach the incompressible limit least squares methods that impose boundary conditions weakly and use an inverse norm are also considered numerical results for a benchmark test problem of planar elasticity are included in order to illustrate the robustness of our

3 regular mixed graphs with optimum hermitian energy abstract let g be a simple undirected graph and g be a mixed graph of g with the generalized orientation and hermitian adjacency matrix h g then g is called the underlying graph of g the hermitian energy of the mixed graph g denoted by e h g is defined as the sum of all the singular values of h g a k regular mixed graph on n vertices having hermitian energy n k is called a k regular optimum hermitian energy mixed graph liu and li 2015 11 proposed the problem of determining all the k regular connected optimum hermitian energy mixed graphs this paper is to give a solution to the problem for the case k 3

anisotropic linear elastic parameter estimation using error in the constitutive equation functional a modified error in the constitutive equation based approach for identification of heterogeneous and linear anisotropic elastic parameters involving static measurements is proposed and explored following an alternating minimization procedure associated with the underlying optimization problem the new strategy results in an explicit material parameter update formula for general anisotropic material this immediately allows us to derive the necessary constraints on measured data and thus restrictions on physical experimentation to achieve the desired reconstruction we consider a few common materials to derive such conditions then we exploit the invariant relationships of the anisotropic constitutive tensor to propose an identification procedure for space dependent material orientations finally we assess the numerical efficacy of the developed tools against a few parameter identification problems of engineering interest

bicyclic oriented graphs with skew rank 2 or 4 the skew rank of oriented graph g denoted by sr g is the rank of the skew adjacency matrix of g the skew rank is even since the skew adjacency matrix is skew symmetric in this paper we characterize the bicyclic oriented graphs with skew rank 2 or 4

on stochastic areas and averages of planar brownian motion duplantier 1989 has computed the characteristic function of the stochastic area of planar brownian motion taken with respect to its centre of gravity in this paper it is shown that this result can be deduced from levy s stochastic area formula moreover duplantier s stochastic area is shown to have the same law as the stochastic area of the restriction to the time interval 0 1 of the brownian ring defined on the time interval 0 4 3 the same method also applies to the stochastic area formulae obtained by biane and yor 1986 which are relative to certain planar processes associated with the orthogonal decomposition of brownian motion along the l2 0 1 basis of legendre polynomials

an equation error approach for the elasticity imaging inverse problem for predicting tumor location the primary objective of this work is to study the elasticity imaging inverse problem of identifying cancerous tumors in the human body this nonlinear inverse problem not only represents an important and interesting application it also brings forth noteworthy mathematical challenges since the underlying model is a system of elasticity equations involving incompressibility due to the locking effect classical finite element methods are not effective for incompressible elasticity equations therefore special treatment is necessary for both the direct and inverse problems to study the inverse problem in an optimization framework we propose an extension of the equation error approach we focus on two cases namely when the material parameter is sufficiently smooth and when it is may be discontinuous for the latter case we extend the total variation regularization method to the elasticity imaging inverse problem we give the existence results for the proposed equation error approach and give the convergence analysis for the discretized problem we give sufficient details on the discrete formulas as well as on the implementation issues numerical examples for smooth and discontinuous coefficients are given

4 regular oriented graphs with optimum skew energy abstract let g be a simple undirected graph and g be an oriented graph of g with the orientation and skew adjacency matrix s g the skew energy of the oriented graph g denoted by e s g is defined as the sum of the absolute values of all the eigenvalues of s g in this paper we characterize the underlying graphs of all 4 regular oriented graphs with optimum skew energy and give orientations of these underlying graphs such that the skew energy of the resultant oriented graphs indeed attain optimum it should be pointed out that there are infinitely many 4 regular connected optimum skew energy oriented graphs while the 3 regular case only has two graphs k 4 the complete graph on 4 vertices and q 3 the hypercube

on the inverse problem of identifying lame coefficients in linear elasticity an output least squares type functional is employed to identify the lame parameters in linear elasticity to be able to identify even the discontinuous lame parameters the regularization is performed by the bv seminorm finite element discretization is used and convergence analysis is given numerical examples are given to show the feasibility of the approach

areas of planar brownian curves addresses the problem of the algebraic area enclosed by a brownian curve in two dimensions recently reconsidered by khandekar and wiegel 1988 the author recalls the principal results actually first obtained by paul levy in 1950 another derivation by functional integrals is given for several probability distributions that of the area of a brownian ring the area between an open brownian arc and its chord and finally the area swept with respect to the mean position of the brownian path as a result which seems to be new

hermitian adjacency matrices and hermitian energies of mixed graphs abstract a complex adjacency matrix of a mixed graph is introduced in the present paper which is a hermitian matrix and called the hermitian adjacency matrix it incorporates both adjacency matrix of an undirected graph and skew adjacency matrix of an oriented graph some of its properties are studied furthermore properties of its characteristic polynomial are studied cospectral problems among mixed graphs including mixed graphs and their underlying graphs oriented graphs and their underlying graphs are studied we give equivalent conditions for a mixed graph especially oriented graph that share the same spectrum with its underlying graph as a consequence we reconfirm a conjecture which was proposed by cui and hou in ref 8 we also show that the spectrum of the hermitian matrix of a mixed graph is invariant when changing the value of any its cut edge if any correspondingly an energy of a mixed graph is introduced and called the hermitian energy it incorporates both the energy of an undirected graph and the skew energy of an oriented graph some of its bounds are given especially the mixed graphs with optimal upper bound of hermitian energy are characterized an infinite family of mixed graphs attaining the maximum hermitian energy is constructed moreover the hermitian energy of a mixed tree is showed to be equal to the energy of its underlying tree finally the integral formula for hermitian energy of a mixed graph is given

on polymer conformations in elongational flows we consider various models of polymer conformations using paths of gaussian processes such as brownian motion in each case the calculation of the law of the moment of inertia of a random polymer structure which is equivalent to the calculation of the partition function is reduced to the problem of finding the law of a certain quadratic functional of a gaussian process we present a new method for computing the laplace transforms of these quadratic functionals which exploit their special form via the ray knight theorem and which does not involve the classical method of eigenvalue expansions we apply the method to several simple examples then show how the same techniques can be applied to more complicated cases with the aid of a little excursion theory

proportionate vs disproportionate distribution of wealth of two individuals in a tempered paretian ensemble abstract we study the distribution p of the random variable x 1 x 1 x 2 where x 1 and x 2 are the wealths of two individuals selected at random from the same tempered paretian ensemble characterized by the distribution x x x 1 where 0 is the pareto index and x is the cut off function we consider two forms of x a bounded function x 1 for l x h and zero otherwise and a smooth exponential function x exp l x x h in both cases x has moments of arbitrary order we show that for 1 p always has a unimodal form and is peaked at 1 2 so that most probably x 1 x 2 for 0 1 we observe a more complicated behavior which depends on the value of l h in particular for c a certain threshold value p has a three modal for a bounded x and a bimodal m shape for an exponential x form which signifies that in such ensembles the wealths x 1 and x 2 are disproportionately different

an inverse problem approach for elasticity imaging through vibroacoustics a methodology for estimating the spatial distribution of elastic moduli using the steady state dynamic response of solids immersed in fluids is presented the technique relies on the ensuing acoustic field from a remotely excited solid to inversely estimate the spatial distribution of young s modulus of biological structures e g breast tissue this work proposes the use of gaussian radial basis functions grbf to represent the spatial variation of elastic moduli grbf are shown to possess the advantage of representing smooth functions with quasi compact support and can efficiently represent elastic moduli distributions such as those that occur in soft biological tissue in the presence of unhealthy tissue e g tumors and calcifications the direct problem consists of a coupled acoustic structure interaction boundary value problem solved in the frequency domain using the finite element method the inverse problem is cast as an optimization problem in which the error functional is defined as a measure of discrepancy between an experimentally measured response and a finite element representation of the system nongradient based optimization algorithms are used to solve the resulting optimization problem the feasibility of the proposed approach is demonstrated through a series of simulations and an experiment for comparison purposes the surface velocity response was also used for the inverse characterization as the measured response in place of the acoustic pressure

on the skew energy of orientations of hypercubes abstract let d be a digraph of order n and 1 2 n denote all the eigenvalues of the skew adjacency matrix of d the skew energy e s d of d is defined as e s d i 1 n i in this paper it is proved that for any positive integer k 3 there exists a k regular graph of order n having an orientation d with e s d n k this work positively answers a problem proposed by adiga et al c adiga r balakrishnan wasin so the skew energy of a digraph linear algebra appl 432 2010 1825 1835 in addition a digraph is also constructed such that its skew energy is the same as the energy of its underlying graph

bicyclic oriented graphs with the second largest skew energy abstract let g be an oriented graph and p s g x det x i s g be the skew characteristic polynomial of its skew adjacency matrix s g the skew energy of g is defined to be the sum of the absolute values of eigenvalues of s g in this paper we firstly find a novel relation between the coefficients of skew characteristic polynomial of unicyclic graph with the third largest skew energy and bicyclic oriented graphs and then use it along with other techniques to characterize the bicyclic oriented graphs with the second largest skew energy

physiologic parameter estimation using inverse problems the estimation of in vivo physiologic parameters is an important but difficult issue in bio medicine therefore the development of mathematical techniques predicting these parameter values is very relevant in a previous work we have proposed a convection diffusion shape model which correlates colonic crypt patterns with the cellular kinetics occurring inside the crypts this correlation is significant in the context of colorectal cancer this model involves several physiologic parameters for which only qualitative information is available in the literature such as the birth rate of proliferative cells in this paper we present a framework for estimating this birth rate parameter in a colonic crypt assuming that the distribution of proliferative cells is known in that crypt more precisely we resolve an inverse problem where the unknown coefficient field the birth rate is connected to the observed measurements the proliferative cell density through a partial differential equation this inver

a levenberg marquardt method based on sobolev gradients abstract we extend the theory of sobolev gradients to include variable metric methods such as newton s method and the levenberg marquardt method as gradient descent iterations associated with stepwise variable inner products in particular we obtain existence uniqueness and asymptotic convergence results for a gradient flow based on a variable inner product