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"Berman, Abraham"
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Conservative port-to-port funneling of light in nonlinear photonic lattices
Photonic systems that steer, guide, or focus light at the microscale are usually designed under passive and lossless conditions. However, Hermiticity, which governs wave evolution in such conservative environments, strictly limits how light can propagate and flow, preventing power from orthogonal inputs to merge into a single coherent channel. Here, we show that this fundamental barrier can be overcome through a nonlinear wave-dynamic process inaccessible under linear Hermitian conditions—the conservative funneling of light. We conceptualize this effect as an all-optical thermodynamic process, whereby wave packets, regardless of origin or coherence, are carried toward the lattice center and combine into a localized ground state. We identify a parametric regime of high powers where kinetic and nonlinear photon energies facilitate the irreversible transport of optical power. Our results demonstrate >70% port-to-port efficiency in a 20-channel system, establishing a robust framework for a universal, fully conservative light funnel.
Light propagation can be controlled via passive—and ideally lossless—photonic systems, where constraints are imposed by the system linearity and Hermiticity. Here, authors describe funnelling of light in a nonlinear Hermitian photonic lattice, achieving a port-to-port funnelling efficiency of 70%.
Journal Article
On Spectral Graph Determination
The study of spectral graph determination is a fascinating area of research in spectral graph theory and algebraic combinatorics. This field focuses on examining the spectral characterization of various classes of graphs, developing methods to construct or distinguish cospectral nonisomorphic graphs, and analyzing the conditions under which a graph’s spectrum uniquely determines its structure. This paper presents an overview of both classical and recent advancements in these topics, along with newly obtained proofs of some existing results, which offer additional insights.
Journal Article
Triangle-free graphs and completely positive matrices
Completely positive matrices are matrices that can be decomposed as BBT, where B is an entrywise nonnegative matrix. These matrices have many applications, including applications to optimization. This article is a survey of some results in the theory of completely positive matrices that involve matrices whose graph contains no triangles.
Journal Article
Completely positive matrices
A real matrix is positive semidefinite if it can be decomposed as A=BB?. In some applications the matrix B has to be elementwise nonnegative. If such a matrix exists, A is called completely positive. The smallest number of columns of a nonnegative matrix B such that A=BB? is known as the cp-rank of A.This invaluable book focuses on necessary conditions and sufficient conditions for complete positivity, as well as bounds for the cp-rank. The methods are combinatorial, geometric and algebraic. The required background on nonnegative matrices, cones, graphs and Schur complements is outlined.
eBook
11 can be reduced to 10
Laffey and Smigoc proved that for every 2x2 doubly nonnegative integer matrix A, icpr(A) is less than or equal to 11. We prove that 11 can be replaced by 10, and show that for many small matrices, even by 9.15
Paper
Heuristic literacy development and its relation to mathematical achievements of middle school students
The relationships between heuristic literacy development and mathematical achievements of middle school students were explored during a 5-month classroom experiment in two 8th grade classes (N = 37). By heuristic literacy we refer to an individual's capacity to use heuristic vocabulary in problem-solving discourse and to approach scholastic mathematical problems by using a variety of heuristics. During the experiment the heuristic constituent of curriculum-determined topics in algebra and geometry was gradually revealed and promoted by means of incorporating heuristic vocabulary in classroom discourse and seizing opportunities to use the same heuristics in different mathematical contexts. Students' heuristic literacy development was indicated by means of individual thinking-aloud interviews and their mathematical achievements — by means of the Scholastic Aptitude Test. We found that heuristic literacy development and changes in mathematical achievements are correlated yet distributed unequally among the students. In particular, the same students, who progressed with respect to SAT scores, progressed also with respect to their heuristic literacy. Those students, who were weaker with respect to SAT scores at the beginning of the intervention, demonstrated more significant progress regarding both measures.
Journal Article
New constructions of non-regular cospectral graphs
We consider two types of joins of graphs \\(G_{1}\\) and \\(G_{2}\\), \\(G_{1}\\veebar G_{2}\\) - the Neighbors Splitting Join and \\(G_{1}\\underset{=}{\\lor}G_{2}\\) - the Non Neighbors Splitting Join, and compute the adjacency characteristic polynomial, the Laplacian characteristic polynomial and the signless Laplacian characteristic polynomial of these joins. When \\(G_{1}\\) and \\(G_{2}\\) are regular, we compute the adjacency spectrum, the Laplacian spectrum, the signless Laplacian spectrum of \\(G_{1}\\underset{=}{\\lor}G_{2}\\) and the normalized Laplacian spectrum of \\(G_{1}\\veebar G_{2}\\) and \\(G_{1}\\underset{=}{\\lor}G_{2}\\). We use these results to construct non regular, non isomorphic graphs that are cospectral with respect to the four matrices: adjacency, Laplacian , signless Laplacian and normalized Laplacian.
Paper
On the transitivity of Gilbert graphs and their complements
The Gilbert graph \\(\\text{Gilbert}(q,n,d)\\), which arises naturally in graph theory and coding theory, is the regular graph on \\(\\mathbb{F}_q^n\\) in which two vertices are adjacent if their Hamming distance is less than \\(d\\), and it is vertex-transitive. We classify all parameters \\((q,n,d)\\) for which \\(\\text{Gilbert}(q,n,d)\\) is edge-transitive or distance-transitive, and separately classify all parameters for which its complement has these properties. We prove that \\(\\text{Gilbert}(q,n,d)\\) is edge-transitive if and only if it is distance-transitive, and that this occurs precisely when \\(d=2\\), \\((q,d)=(2,3)\\), or \\((q,d)=(2,n)\\). For the complement graphs, we determine all parameters yielding edge- or distance-transitivity using spectral methods based on Krawtchouk polynomials and the structure of the Hamming association scheme. In contrast to the Gilbert graphs, where the parameter sets corresponding to edge- and distance-transitivity coincide, we show that for their complements the set of parameters yielding distance-transitivity is strictly contained in the set yielding edge-transitivity. As an application, we compute the exact values of the Lov\\'{a}sz \\(\\vartheta\\)-function of Gilbert graphs, as well as of their complements, in all cases where either one of them is edge-transitive.
Paper
When Do Gifted High School Students Use Geometry to Solve Geometry Problems?
This article describes the following phenomenon: Gifted high school students trained in solving Olympiad-style mathematics problems experienced conflict between their conceptions of effectiveness and elegance (the EEC). This phenomenon was observed while analyzing clinical task-based interviews that were conducted with three members of the Israeli team participating in the International Mathematics Olympiad. We illustrate how the conflict between the students’ conceptions of effectiveness and elegance is reflected in their geometrical problem solving, and analyze didactical and epistemological roots of the phenomenon.
Journal Article