3D Structural Model and Visualization of Blood Vessels Based on L-System

The insight in structures of the blood vessels is a basis for study of blood flows to help understanding the abnormalities of blood vessels that can cause vascular diseases. Basic concept used for constructing structures of blood vessels in organs is arterial branching, which is usually characterized by fractal similarity in the bifurcation pattern. In this work, the concept of Lindenmayer system (L-system) is modified for three-dimensional (3D) tree-like structures to model structures of blood vessels in organs, and then, applied to construct and visualize structural blood vessels via our software created based on openGL and Lazarus program. The structure of blood vessels is constructed based on the physiological law of arterial branching proposed Murray (Murray’s law) under additional assumptions and constraints such as the spreading of blood vessels to cover all directions, the angle condition and the non-overlapping vessels condition. The concept is applied to simulate structures of blood vessels in 3 study cases, including symmetric arterial branching, non-symmetric arterial branching and structure of blood vessel on different domains. The results of structures of blood vessels generated from all cases are measured based on the number of segments, the total blood volume and the fractal dimension. The results of modeling and simulation in this work are illustrated by comparing with other results appeared literature. Moreover, the constructed structures of the blood vessels based on this 3D L-system could be useful for future research such as blood flow, pressure and other properties involving in structures of blood vessels in different organs of human and animals. HIGHLIGHTS A new 3D L-system is developed based on directional vectors for construction of 3D tree-like structures such as structures of blood vessels The model of structures of blood vessels is constructed based on the physiological laws of arterial branching (Murray’s law) with additional assumptions on the spreading of blood vessels, the angle condition, and the non-overlapping of blood vessels Algorithm and software are developed based on L-system to simulate and visualize 3D structures of blood vessels GRAPHICAL ABSTRACT

Fano schemes of complete intersections in toric varieties

We study Fano schemes $$\mathrm{F}_k(X)$$ F k ( X ) for complete intersections X in a projective toric variety $$Y\subset \mathbb {P}^n$$ Y ⊂ P n . Our strategy is to decompose $$\mathrm{F}_k(X)$$ F k ( X ) into closed subschemes based on the irreducible decomposition of $$\mathrm{F}_k(Y)$$ F k ( Y ) as studied by Ilten and Zotine. We define the “expected dimension” for these subschemes, which always gives a lower bound on the actual dimension. Under additional assumptions, we show that these subschemes are non-empty and smooth of the expected dimension. Using tools from intersection theory, we can apply these results to count the number of linear subspaces in X when the expected dimension of $$\mathrm{F}_k(X)$$ F k ( X ) is zero.

Distortion inequality for a Markov operator generated by a randomly perturbed family of Markov Maps in ℝ d

Asymptotic properties of the sequences (a) { P j } j = 1 ∞ $\{P^{j}\}_{j=1}^{\infty}$ and (b) { j − 1 ∑ i = 0 j − 1 P i } j = 1 ∞ $\{ j^{-1} \sum _{i=0}^{j-1} P^{i}\}_{j=1}^{\infty}$ are studied for g ∈ G = {f ∈ L 1(I) : f ≥ 0 and ‖f ‖ = 1}, where P : L 1(I) → L 1(I) is a Markov operator defined by P f := ∫ P y f d p ( y ) $Pf:= \int P_{y}f\, dp(y) $ for f ∈ L 1; {Py } y∈Y is the family of the Frobenius-Perron operators associated with a family {φy } y∈Y of nonsingular Markov maps defined on a subset I ⊆ ℝ d ; and the index y runs over a probability space (Y, Σ(Y), p). Asymptotic properties of the sequences (a) and (b), of the Markov operator P, are closely connected with the asymptotic properties of the sequence of random vectors x j = φ ξ j ( x j − 1 ) $x_{j}=\varphi_{\xi_{j}}(x_{j-1})$ for j = 1,2, . . .,where { ξ j } j = 1 ∞ $\{\xi_{j}\}_{j=1}^{\infty}$ is a sequence of Y-valued independent random elements with common probability distribution p. An operator-theoretic analogue of Rényi’s Condition is introduced for the family {Py } y∈Y of the Frobenius-Perron operators. It is proved that under some additional assumptions this condition implies the L 1- convergence of the sequences (a) and (b) to a unique g 0 ∈ G. The general result is applied to some families {φy } y∈Y of smooth Markov maps in ℝ d .

Online Premeans and Their Computation Complexity

We extend some approach to the family of symmetric means (i.e. symmetric functions $$\mathscr {M} :\bigcup _{n=1}^\infty I^n \rightarrow I$$ M : ⋃ n = 1 ∞ I n → I with $$\min \le \mathscr {M}\le \max $$ min ≤ M ≤ max ; I is an interval). Namely, it is known that every symmetric mean can be written in a form $$\mathscr {M}(v_1,\dots ,v_n):=F(f(v_1)+\cdots +f(v_n))$$ M ( v 1 , ⋯ , v n ) : = F ( f ( v 1 ) + ⋯ + f ( v n ) ) , where $$f :I \rightarrow G$$ f : I → G and $$F :G \rightarrow I$$ F : G → I (G is a commutative semigroup). For $$G=\mathbb {R}^k$$ G = R k or $$G=\mathbb {R}^k \times \mathbb {Z}$$ G = R k × Z ($$k \in \mathbb {N}$$ k ∈ N ) and continuous functions f and F we obtain two series of families (depending on k). It can be treated as a measure of complexity in a family of means (this idea is inspired by theory of regular languages and algorithmics). As a result we characterize the celebrated families of quasi-arithmetic means ($$G=\mathbb {R}\times \mathbb {Z}$$ G = R × Z ) and Bajraktarević means ($$G=\mathbb {R}^2$$ G = R 2 under some additional assumptions). Moreover, we establish certain estimations of complexity for several other classical families.

JORDAN–KRONECKER INVARIANTS OF LIE ALGEBRA REPRESENTATIONS AND DEGREES OF INVARIANT POLYNOMIALS

For an arbitrary representation ρ of a complex finite-dimensional Lie algebra, we construct a collection of numbers that we call the Jordan–Kronecker invariants of ρ. Among other interesting properties, these numbers provide lower bounds for degrees of polynomial invariants of ρ. Furthermore, we prove that these lower bounds are exact if and only if the invariants are independent outside of a set of large codimension. Finally, we show that under certain additional assumptions our bounds are exact if and only if the algebra of invariants is freely generated.

Anisotropy Measure from Three Diffusion-Encoding Gradient Directions

PurposeWe propose a method that can provide information about the anisotropy and orientation of diffusion in the brain from only 3 orthogonal gradient directions without imposing additional assumptions.MethodsThe method is based on the Diffusion Anisotropy (DiA) that measures the distance from a diffusion signal to its isotropic equivalent. The original formulation based on a Spherical Harmonics basis allows to go down to only 3 orthogonal directions in order to estimate the measure. In addition, an alternative simplification and a color-coding representation are also proposed.ResultsAcquisitions from a publicly available database are used to test the viability of the proposal. The DiA succeeded in providing anisotropy information from the white matter using only 3 diffusion-encoding directions. The price to pay for such reduced acquisition is an increment in the variability of the data and a subestimation of the metric.ConclusionsThe calculation of anisotropy information from DMRI is feasible using fewer than 6 gradient directions by using DiA. The method is totally compatible with existing acquisition protocols and it may provide complementary information about orientation in fast diffusion acquisitions.

Estimating and Using Individual Marginal Component Effects from Conjoint Experiments

Conjoint experiments are quickly gaining popularity as a vehicle for studying multidimensional political preferences. A common way to explore heterogeneity of preferences estimated with conjoint experiments is by estimating average marginal component effects across subgroups. However, this method does not give the researcher the full access to the variation of preferences in the studied populations, as that would require estimating effects on the individual level. Currently, there is no accepted technique to obtain estimates of individual-level preferences from conjoint experiments. The present paper addresses this gap by proposing a procedure to estimate individual preferences as respondent-specific marginal component effects. The proposed strategy does not require any additional assumptions compared to the standard conjoint analysis, although some changes to the task design are recommended. Methods to account for uncertainty in resulting estimates are also discussed. Using the proposed procedure, I partially replicate a conjoint experiment on immigrant admission with recommended design adjustments. Then, I demonstrate how individual marginal component effects can be used to explore distributions of preferences, intercorrelations between different preference dimensions, and relationships of preferences to other variables of interest.

Nonuniqueness and Uniqueness for Inverse Magnetization Problems

<p>Nonuniqueness is a well-known issue with inverse problems involving geophysical potential fields (typically gravitational or magnetic fields). If no additional assumptions are made on the underlying source, only certain harmonic contributions can be reconstructed uniquely from knowledge of the potential. Such harmonic contributions have no intuitive geophysical interpretation. However, in various applications some specific properties are of particular interest: e.g., the direction of the magnetization in paleomagnetic studies or the lithospheric susceptibility in geomagnetism. In this presentation, we give a brief overview on the characterization of nonuniqueness and on a priori assumptions on the underlying magnetization that might lead to uniqueness or at least partial uniqueness.</p>

Quantum prescriptions are more ontologically distinct than they are operationally distinguishable

Based on an intuitive generalization of the Leibniz principle of `the identity of indiscernibles', we introduce a novel ontological notion of classicality, called bounded ontological distinctness. Formulated as a principle, bounded ontological distinctness equates the distinguishability of a set of operational physical entities to the distinctness of their ontological counterparts. Employing three instances of two-dimensional quantum preparations, we demonstrate the violation of bounded ontological distinctness or excess ontological distinctness of quantum preparations, without invoking any additional assumptions. Moreover, our methodology enables the inference of tight lower bounds on the extent of excess ontological distinctness of quantum preparations. Similarly, we demonstrate excess ontological distinctness of quantum transformations, using three two-dimensional unitary transformations. However, to demonstrate excess ontological distinctness of quantum measurements, an additional assumption such as outcome determinism or bounded ontological distinctness of preparations is required. Moreover, we show that quantum violations of other well-known ontological principles implicate quantum excess ontological distinctness. Finally, to showcase the operational vitality of excess ontological distinctness, we introduce two distinct classes of communication tasks powered by excess ontological distinctness.

Strongly Peaking Representations and Compressions of Operator Systems

We use Arveson’s notion of strongly peaking representation to generalize uniqueness theorems for free spectrahedra and matrix convex sets that admit minimal presentations. A fully compressed separable operator system necessarily generates the $C^*$-envelope and is such that the identity is the direct sum of strongly peaking representations. In particular, a fully compressed presentation of a separable operator system is unique up to unitary equivalence. Under various additional assumptions, minimality conditions are sufficient to determine a separable operator system uniquely.