Infinite Pairs Research Articles

Existence and multiplicity of solutions for a Steklov eigenvalue problem involving the p(x)-Laplacian-like operator

Using the variational method, we prove theexistence and multiplicity of solutions for a Steklov problem involving the $p(x)$-Laplacian-like operator, originated from a capillary phenomena. Especially, an existence criterion for infinite many pairs of solutions for the problem is obtained.

On the eigenvalues and Seidel eigenvalues of chain graphs

In this paper, we primarily focus on the eigenvalues of the adjacency matrix and Seidel matrix of chain graphs, referred to as eigenvalues and Seidel eigenvalues of these graphs, respectively. Firstly, we utilize the characteristic polynomial of the adjacency matrix of a chain graph to construct infinite pairs of non-isomorphic cospectral chain graphs. Next, we determine the inertia of the Seidel matrix of a chain graph and establish an interval that does not contain the Seidel eigenvalues of chain graphs. Lastly, we characterize chain graphs with distinct Seidel eigenvalues.

On reconstructing finite gauge group from fusion rules

Gauging a finite group 0-form symmetry G of a quantum field theory (QFT) results in a QFT with a Rep(G) symmetry implemented by Wilson lines. The group G determines the fusion of Wilson lines. However, in general, the fusion rules of Wilson lines do not determine G. In this paper, we study the properties of G that can be determined from the fusion rules of Wilson lines and surface operators obtained from higher-gauging Wilson lines. This is in the spirit of Richard Brauer who asked what information in addition to the character table of a finite group needs to be known to determine the group. We show that fusion rules of surface operators obtained from higher-gauging Wilson lines can be used to distinguish infinite pairs of groups which cannot be distinguished using the fusion of Wilson lines. We derive necessary conditions for two non-isomorphic groups to have the same surface operator fusion and find a pair of such groups.

Construction of Equienergetic Trees

The energy E(G) of a graph G is the sum of the absolute values of all eigenvalues of G. Two graphs of the same order are said to be equienergetic if their energies are equal. As pointed out by Gutman, it is not known how to systematically construct any pair of equienergetic, non-cospectral trees until now. Inspired by the research of integral trees, we proposed a construction of infinite pairs of equienergetic trees of diameter 4.

Discriminating States of Polarization

Equiprobable incoherent mixtures of two totally polarized states of light whose associated three-dimensional Jones vectors are mutually orthogonal are called discriminating states and constitute a peculiar type of state that plays a key role in the characteristic decomposition of a generic state into a totally polarized state, a totally unpolarized state, and a discriminating state. In general, discriminating states are three-dimensional, in the sense that the strengths of the three components of the electric field are nonzero for any Cartesian reference frame considered. In the limiting case that the electric field evolves in a fixed plane, the discriminating state is said to be regular and corresponds to a two-dimensional unpolarized state. The special features of discriminating states cover, e.g., their possible synthesis from infinite pairs of mutually orthogonal states as well as their transverse spin. The nature and properties of discriminating states are comprehensively analyzed based on their associated intrinsic Stokes parameters, which leads to meaningful interpretations in terms of the associated polarization ellipsoids and spin vectors.

Generalized Paley graphs equienergetic with their complements

We consider generalized Paley graphs , generalized Paley sum graphs , and their corresponding complements and , for k = 3, 4. Denote by either or . We compute the spectra of and and from them we obtain the spectra of and also. Then we show that, in the non-semiprimitive case, the spectrum of and with p prime can be recursively obtained, under certain arithmetic conditions, from the spectrum of the graphs and for any , respectively. Using the spectra of these graphs we give necessary and sufficient conditions on the spectrum of such that and are equienergetic for k = 3, 4. In a previous work we have classified all bipartite regular graphs and all strongly regular graphs which are complementary equienergetic, i.e. and are equienergetic pairs of graphs. Here we construct infinite pairs of equienergetic non-isospectral regular graphs which are neither bipartite nor strongly regular.

Spectra of M-bicone product of graphs

In this paper, we introduce a graph product, namely, [Formula: see text]-bicone product and determine the generalized characteristic polynomial of the graphs obtained from this product. Consequently, we obtain the characteristic polynomial of the adjacency matrix, the Laplacian matrix and the signless Laplacian matrix of this graph. Also, from these results, we obtain the [Formula: see text]-spectra of some families of bicone product of graphs. As an application, we obtain infinite pairs of [Formula: see text]-cospectral graphs.

Blind image deconvolution via salient edge selection and mean curvature regularization

Blind image deconvolution is an ill-posed problem since there exists infinite pairs of blur kernels and latent images. To obtain reasonable results of this problem, most previous methods have emphasized the importance of selecting salient edges for blur kernel estimation. In this paper, a blind deconvolution method based on explicit and implicit salient edges selection is proposed. Explicit edge selection is achieved by using mutually guided image filtering, while mean curvature regularization is adopted to remove disadvantageous structures implicitly. Besides, the proposed model can be efficiently optimized by using half-quadratic penalty method and mean curvature filter. Extensive experiments show that the proposed method performs favorably against the state-of-the-art blind deconvolution methods on both synthetic and real-world blurred images.

Infinite co-minimal pairs involving lacunary sequences and generalisations to higher dimensions

The study of minimal complements in a group or a semigroup was initiated by Nathanson. The notion of minimal complements and being a minimal complement leads to the notion of co-minimal pairs which was considered in a prior work of the authors. In this article, we study which type of subsets in the integers and free abelian groups of higher rank can be a part of a co-minimal pair. We show that a majority of lacunary sequences have this property. From the conditions established, one can show that any infinite subset of any finitely generated abelian group has uncountably many subsets which is a part of a co-minimal pair. Further, the uncountable collection of sets can be chosen so that they satisfy certain algebraic properties.

Invariant descriptors for intrinsic reflectance optimization.

Intrinsic image decomposition aims to factorize an image into albedo (reflectance) and shading (illumination) sub-components. Being ill posed and under-constrained, it is a very challenging computer vision problem. There are infinite pairs of reflectance and shading images that can reconstruct the same input. To address the problem, Intrinsic Images in the Wild by Bell et al. provides an optimization framework based on a dense conditional random field (CRF) formulation that considers long-range material relations. We improve upon their model by introducing illumination invariant image descriptors: color ratios. The color ratios and the intrinsic reflectance are both invariant to illumination and thus are highly correlated. Through detailed experiments, we provide ways to inject the color ratios into the dense CRF optimization. Our approach is physics based and learning free and leads to more accurate and robust reflectance decompositions.

Spectra of bowtie product of graphs

In this paper, we introduce a new graph product, namely, bowtie product. We determine the generalized characteristic polynomial of the bowtie product of graphs and consequently, we obtain the characteristic polynomial of the adjacency martix, the Laplacian matrix, the signless Laplacian matrix and the normalized Laplacian matrix of this graph. Also, from these results, we obtain the [Formula: see text]-spectra of some families of bowtie product of graphs. As an application, we obtain infinite pairs of [Formula: see text]-cospectral graphs and construct infinite number of [Formula: see text]-integral graphs.

Approaching infinite affinity through engineering of peptide–protein interaction

Much of life's complexity depends upon contacts between proteins with precise affinity and specificity. The successful application of engineered proteins often depends on high-stability binding to their target. In recent years, various approaches have enabled proteins to form irreversible covalent interactions with protein targets. However, the rate of such reactions is a major limitation to their use. Infinite affinity refers to the ideal where such covalent interaction occurs at the diffusion limit. Prototypes of infinite affinity pairs have been achieved using nonnatural reactive groups. After library-based evolution and rational design, here we establish a peptide-protein pair composed of the regular 20 amino acids that link together through an amide bond at a rate approaching the diffusion limit. Reaction occurs in a few minutes with both partners at low nanomolar concentration. Stopped flow fluorimetry illuminated the conformational dynamics involved in docking and reaction. Hydrogen-deuterium exchange mass spectrometry gave insight into the conformational flexibility of this split protein and the process of enhancing its reaction rate. We applied this reactive pair for specific labeling of a plasma membrane target in 1 min on live mammalian cells. Sensitive and specific detection was also confirmed by Western blot in a range of model organisms. The peptide-protein pair allowed reconstitution of a critical mechanotransmitter in the cytosol of mammalian cells, restoring cell adhesion and migration. This simple genetic encoding for rapid irreversible reaction should provide diverse opportunities to enhance protein function by rapid detection, stable anchoring, and multiplexing of protein functionality.

The spectra of a new join of graphs

Let [Formula: see text] and [Formula: see text] be three graphs on disjoint sets of vertices and [Formula: see text] has [Formula: see text] edges. Let [Formula: see text] be the graph obtained from [Formula: see text] and [Formula: see text] in the following way: (1) Delete all the edges of [Formula: see text] and consider [Formula: see text] disjoint copies of [Formula: see text]. (2) Join each vertex of the [Formula: see text]th copy of [Formula: see text] to the end vertices of the [Formula: see text]th edge of [Formula: see text]. Let [Formula: see text] be the graph obtained from [Formula: see text] by joining each vertex of [Formula: see text] with each vertex of [Formula: see text] In this paper, we determine the adjacency (respectively, Laplacian, signless Laplacian) spectrum of [Formula: see text] in terms of those of [Formula: see text] and [Formula: see text] As an application, we construct infinite pairs of cospectral graphs.

On the Wiener index, distance cospectrality and transmission-regular graphs

In this paper, we investigate various algebraic and graph theoretic properties of the distance matrix of a graph. Two graphs are D-cospectral if their distance matrices have the same spectrum. We construct infinite pairs of D-cospectral graphs with different diameter and different Wiener index. A graph is k-transmission-regular if its distance matrix has constant row sum equal to k. We establish tight upper and lower bounds for the row sum of a k-transmission-regular graph in terms of the number of vertices of the graph. Finally, we determine the Wiener index and its complexity for linear k-trees, and obtain a closed form for the Wiener index of block-clique graphs in terms of the Laplacian eigenvalues of the graph. The latter leads to a generalization of a result for trees which was proved independently by Mohar and Merris.

On Global Attractors for a Class of Reaction-Diffusion Equations on Unbounded Domains with Some Strongly Nonlinear Weighted Term

We consider the existence and properties of the global attractor for a class of reaction-diffusion equation∂u/∂t-Δu-u+κ(x)|u|p-2u+f(u)=0, in Rn×R+; u(x,0)=u0(x), in Rn. Under some suitable assumptions, we first prove that the problem has a global attractorAinL2(Rn). Then, by using theZ2-index theory, we verify thatAis an infinite dimensional set and it contains infinite distinct pairs of equilibrium points.

Adjoining to [formula omitted]-Wythoff’s game its [formula omitted]-positions as moves

We define a class of new games obtained from (s,t)-Wythoff’s game by adjoining to it an appropriate subset of its P-positions as additional moves. The preservation of all P-positions of the new games are analyzed. Our results show that for any integer s≥1, there exist infinite pairs of (s,t) such that all P-positions of the new games do not depend on the choice of additional moves.

Multiplicity of Solutions on a Nonlinear Eigenvalue Problem for p(x)-Laplacian-like Operators

The paper study the existence and multiplicity of solutions for the nonlinear eigenvalue problems for p(x)-Laplacian-like operators, originated from a capillary phenomena. Especially, an existence criterion for infinite many pairs of solutions for the problem is obtained.

Infinite Lie algebras and dual pairs in 4D CFT models

It is known that there are no local scalar Lie fields in more than two dimensions. Bilocal fields, however, which naturally arise in conformal operator product expansions, do generate infinite Lie algebras. It is demonstrated that these Lie algebras of local observables admit (highly reducible) unitary positive energy representations in a Fock space. The multiplicity of their irreducible components is governed by a compact gauge group. The mutually commuting observable algebra and gauge group form a dual pair in the sense of Howe. In a theory of local scalar fields of conformal dimension two in four space-time dimensions the associated dual pairs are constructed and classified. The talk reviews joint work of B. Bakalov, N. M. Nikolov, K.-H. Rehren, and the author.

Existence of solutions for [formula omitted]-Laplacian equations with singular coefficients in [formula omitted

In this paper, we deal with the existence of solutions for the following p ( x ) -Laplacian equations via critical point theory { − div ( | ∇ u | p ( x ) − 2 ∇ u ) + e ( x ) | u | p ( x ) − 2 u = f ( x , u ) in R N , u ∈ W 1 , p ( x ) ( R N ) , where f ( x , u ) = ∑ i = 1 m λ i a i ( x ) g i ( x , u ) , g i : R N × R → R satisfies the Caratheodory condition, but a i ( x ) are singular. Especially, we obtain existence criterion for infinite many pairs of solutions for the problem, when some a i 0 ( x ) can change sign and g i 0 ( x , ⋅ ) satisfies super- p + growth condition.

Inverse gas chromatography study on the effect of humidity on the mass transport of alcohols in an ethylene-vinyl alcohol copolymer near the glass transition temperature

Inverse gas chromatography (IGC) was used to study the effect of moisture on transport properties of three low molecular weight alcohols (methanol, ethanol, and 1-butanol) through high barrier copolymers of ethylene-vinyl alcohol with an ethylene content of 38% mol (EVOH38) at 40 °C. The value of the partition coefficient ( K) was obtained by using two approaches: (a) the fit of the slope of sorption isotherms obtained through the method of Kiselev and Yashin; and (b) the solution to the model of Romdhane and Danner obtained by using the law of moments. The second method also allowed the estimation of the diffusion coefficient ( D p) at the different humidity conditions. None of these two methods were applicable at low values of relative humidity. With the first method, the diffusion of the permeants through the copolymer was not fast enough to allow them to reach the core of the EVOH particles used as stationary phase resulting in sorption values unrealistically low. The fit of the chromatograms obtained by using the second method also suggested questionable values of the mass transport parameters. Although the theoretical curve perfectly described the chromatogram, the low extent of the interaction and the slow diffusion resulted in interdependent values of the coefficients K and D p, with infinite pairs of values providing the same curve profile. As the relative humidity of the carrier gas increased, the diffusivity and the sorption of the alcohols also increased, making both methods applicable. In the case of the partition coefficient, the sorption of the biggest molecules (ethanol and 1-butanol) was the most affected, the increment of K for methanol being moderate. As regards the D p value, methanol was the most influenced compound and 1-butanol the least. Finally, a sharp increment of the D p of the three alcohols was observed between 35 and 47% RH and attributed to the plasticization of the copolymer.