Symmetric Self Research Articles

On blow up for the energy super critical defocusing nonlinear Schr\xf6dinger equations

We consider the energy supercritical defocusing nonlinear Schrödinger equation i∂tu+Δu-u|u|p-1=0\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} i\\partial _tu+\\Delta u-u|u|^{p-1}=0 \\end{aligned}$$\\end{document}in dimension dge 5. In a suitable range of energy supercritical parameters (d, p), we prove the existence of {mathcal {C}}^infty well localized spherically symmetric initial data such that the corresponding unique strong solution blows up in finite time. Unlike other known blow up mechanisms, the singularity formation does not occur by concentration of a soliton or through a self similar solution, which are unknown in the defocusing case, but via a front mechanism. Blow up is achieved by compression for the associated hydrodynamical flow which in turn produces a highly oscillatory singularity. The front blow up profile is chosen among the countable family of {mathcal {C}}^infty spherically symmetric self similar solutions to the compressible Euler equation whose existence and properties in a suitable range of parameters are established in the companion paper (Merle et al. in Preprint (2019)) under a non degeneracy condition which is checked numerically.

Compactness and Finiteness Theorems for Rotationally Symmetric Self Shrinkers

In this note, we first show a compactness theorem for rotationally symmetric self shrinkers of entropy less than 2, concluding that there are entropy minimizing self shrinkers diffeomorphic to $$S^1 \times S^{n-1}$$ for each $$n \ge 2$$ in the class of rotationally symmetric self shrinkers. Assuming extra symmetry, namely that the profile curve is convex, we remove the entropy assumption. Supposing the profile curve is additionally reflection symmetric we show there are only finitely many such shrinkers up to rigid motion.

On the Stability of Type I Blow Up For the Energy Super Critical Heat Equation

We consider the energy super critical semilinear heat equation ∂tu = ∆u + u p , x ∈ R 3 , p > 5. We first revisit the construction of radially symmetric self similar solutions performed through an ode approach in [51], [2], and propose a bifurcation type argument suggested in [3] which allows for a sharp control of the spectrum of the corresponding linearized operator in suitable weighted spaces. We then show how the sole knowledge of this spectral gap in weighted spaces implies the finite codimensional non radial stability of these solutions for smooth well localized initial data using energy bounds. The whole scheme draws a route map for the derivation of the existence and stability of self similar blow up in non radial energy super critical settings.

Definition of complexity for dynamical spherically symmetric dissipative self-gravitating fluid distributions

The recently proposed definition of complexity for static and spherically symmetric self--gravitating systems [1], is extended to the fully dynamic situation. In this latter case we have to consider not only the complexity factor of the structure of the fluid distribution, but also the condition of minimal complexity of the pattern of evolution. As we shall see these two issues are deeply intertwined. For the complexity factor of the structure we choose the same as for the static case, whereas for the simplest pattern of evolution we assume the homologous condition. The dissipative and non-dissipative cases are considered separately. In the latter case the fluid distribution, satisfying the vanishing complexity factor condition and evolving homologously, corresponds to a homogeneous (in the energy density), geodesic and shear-free, isotropic (in the pressure) fluid. In the dissipative case the fluid is still geodesic, but shearing, and there exists (in principle) a large class of solutions. Finally we discuss about the stability of the vanishing complexity condition.

New definition of complexity for self-gravitating fluid distributions: The spherically symmetric, static case

We put forward a new definition of complexity, for static and spherically symmetric self--gravitating systems, based on a quantity, hereafter referred to as complexity factor, that appears in the orthogonal splitting of the Riemann tensor, in the context of general relativity. We start by assuming that the homogeneous (in the energy density) fluid, with isotropic pressure is endowed with minimal complexity. For this kind of fluid distribution, the value of complexity factor is zero. So, the rationale behind our proposal for the definition of complexity factor stems from the fact that it measures the departure, in the value of the active gravitational mass (Tolman mass), with respect to its value for a zero complexity system. Such departure is produced by a specific combination of energy density inhomogeneity and pressure anisotropy. Thus, zero complexity factor may also be found in self--gravitating systems with inhomogeneous energy density and anisotropic pressure, provided the effects of these two factors, on the complexity factor, cancel each other. Some exact interior solutions to the Einstein equations satisfying the zero complexity criterium are found, and prospective applications of this newly defined concept, to the study of the structure and evolution of compact objects, are discussed.

Gog and GOGAm pentagons

We consider the problem of finding a bijection between the sets of alternating sign matrices and of totally symmetric self complementary plane partitions, which can be reformulated using Gog and Magog triangles. In a previous work we introduced GOGAm triangles, which are images of Magog triangles by the Schützenberger involution. In this paper we introduce Gog and GOGAm pentagons. We conjecture that they are equienumerated. We provide some numerical evidence as well as an explicit bijection in the case when they have one or two diagonals. We also consider some interesting statistics on Gog and Magog triangles.

Stability of Charged Shell Supported by a Phantom Energy

This paper discusses the evolution of a thin spherically symmetric self gravitating phantom shell around the charged shell. The general equations describing the motion of shell with a general form of equation of state are derived. The different types of space-time R± and T± regions and shell motion are classified depending on the parameters of the problem. The mechanical stability analysis of this spherically symmetric thin shell with charge in Reissner- Nordstrom (RN) to linearized spherically symmetric perturbation about static equilibrium solution is carried out.

Gravity effects on thick brane formation from scalar field dynamics

The formation of a thick brane in five-dimen\-sional space-time is investigated when warp geometries of $AdS_5$ type are induced by scalar matter dynamics and triggered by a thin-brane defect. The scalar matter is taken to consist of two fields with $O(2)$ symmetric self interaction and with manifest $O(2)$ symmetry breaking by terms quadratic in fields. One of them serves as a thick brane formation mode around a kink background and another one is of a Higgs-field type which may develop a classical background as well. Scalar matter interacts with gravity in the minimal form and gravity effects on (quasi)localized scalar fluctuations are calculated with usage of gauge invariant variables suitable for perturbation expansion. The calculations are performed in the vicinity of the critical point of spontaneous breaking of the combined parity symmetry where a non-trivial v.e.v. of the Higgs-type scalar field is generated. The nonperturbative discontinuous gravitational effects in the mass spectrum of light localized scalar states are studied in the presence of a thin-brane defect. The thin brane with negative tension happens to be the most curious case when the singular barriers form a potential well with two infinitely tall walls and the discrete spectrum of localized states arises completely isolated from the bulk.

EMBEDDING PROPERTIES IN NEAR-RINGS

In this paper, we initiate the study of zero symmetric and constant parts of near-rings, and then apply these to self map near-rings. Next, we investigate that every near-ring can be embedded into some self map near-ring, and every zero symmetric near-ring can be embedded into some zero symmetric self map near-ring.

Localization of scalar fields on self-gravitating thick branes

The model of a domain wall ("thick" brane) in noncompact five-dimensional space-time is considered with geometries of $AdS_5$ type generated by self-interacting scalar matter. The scalar matter is composed of two fields with O(2) symmetric self interaction. One of them is mixed with gravity scalar modes and plays role of the brane formation mode (due to a kink background) and another one is of a Higgs-field type. The interplay between soft breaking of O(2) symmetry and gravity influence is thoroughly investigated around the critical point of spontaneous $\tau$ symmetry breaking when the v.e.v. of the Higgs-type scalar field occurs. The possibility of (quasi)localization of scalar modes on such thick branes is examined.

Charged shell supported by a phantom energy

This paper discusses the evolution of a thin spherically symmetric self gravitating phantom shell around the charged shell. The general equations describing the motion of shell with a general form of equation of state are derived. The different types of space-time R± and T± regions and shell motion are classified depending on the parameters of the problem.

On Zeilberger's Constant Term for Andrews' TSSCPP Theorem

This paper studies Zeilberger's two prized constant term identities. For one of the identities, Zeilberger asked for a simple proof that may give rise to a simple proof of Andrews theorem for the number of totally symmetric self complementary plane partitions. We obtain an identity reducing a constant term in $2k$ variables to a constant term in $k$ variables. As applications, Zeilberger's constant terms are converted to single determinants. The result extends for two classes of matrices, the sum of all of whose full rank minors is converted to a single determinant. One of the prized constant term problems is solved, and we give a seemingly new approach to Macdonald's constant term for root system of type BC.

The poset perspective on alternating sign matrices

Alternating sign matrices (ASMs) are square matrices with entries 0, 1, or -1 whose rows and columns sum to 1 and whose nonzero entries alternate in sign. We put ASMs into a larger context by studying the order ideals of subposets of a certain poset, proving that they are in bijection with a variety of interesting combinatorial objects, including ASMs, totally symmetric self―complementary plane partitions (TSSCPPs), Catalan objects, tournaments, semistandard Young tableaux, and totally symmetric plane partitions. We use this perspective to prove an expansion of the tournament generating function as a sum over TSSCPPs which is analogous to a known formula involving ASMs. Les matrices à signe alternant (ASMs) sont des matrices carrées dont les coefficients sont 0,1 ou -1, telles que dans chaque ligne et chaque colonne la somme des entrées vaut 1 et les entrées non nulles ont des signes qui alternent. Nous incluons les ASMs dans un cadre plus vaste, en étudiant les idéaux des sous-posets d'un certain poset, dont nous prouvons qu'ils sont en bijection avec de nombreux objets combinatoires intéressants, tels que les ASMs, les partitions planes totalement symétriques autocomplémentaires (TSSCPPs), des objets comptés par les nombres de Catalan, les tournois, les tableaux semistandards, ou les partitions planes totalement symétriques. Nous utilisons ce point de vue pour démontrer un développement de la série génératrice des tournois en une somme portant sur les TSSCPPs, analogue à une formule déjà connue faisant appara\^ıtre les ASMs.

On the Doubly Refined Enumeration of Alternating Sign Matrices and Totally Symmetric Self - Complementary Plane Partitions

We prove the equality of doubly refined enumerations of Alternating Sign Matrices and of Totally Symmetric Self-Complementary Plane Partitions using integral formulae originating from certain solutions of the quantum Knizhnik–Zamolodchikov equation.

Anisotropic fluid distribution in higher dimensional general theory of relativity

We have obtained static and spherically symmetric self-gravitating solution of the field equations for anisotropic distribution of matter in higher- dimensional in the context of Einstein’s general theory of relativity. This work is an extension of the previous work of Hector Rago (Astrophys. Space Sci. 183:333, 1991) for four dimensional space-time. The solutions are matched to the analytical solutions for spherically symmetric self gravitating distribution of anisotropic matter obtained by Hector Rago (1991) for n=2.

Visualization of Host–Guest Interactions Between a Modified Nucleoside and Organic Moieties. Crystal Structures of 2'-Deoxy 5'-O-trityluridine and 2'-Deoxy 5'-O-tritylthymidine with Benzene, Toluene, Xylene, Trimethylbenzene, Cyclohexane and Water

This paper describes the crystal structures of 2’-deoxy 5’-O-trityluridine(5’-TU) and20-deoxy 5’-O-tritylthymidine (5’-TT) containing different organic moieties. There are two crystallographically independent nucleoside molecules present in the asymmetric units of all the structures. Uracil and thymine bases of the 2’-deoxy 5’-trityl uridine 5’-TU) and all the 2’-deoxy 5’- tritylthymidine structures are in an anti conformation with respect to their furanose rings. Thymine bases of molecules A and B form symmetric self pairs through N(3)—O(2) hydrogen bonds, whereas uracil bases are not engaged in hydrogen bonding between themselves. Ribose moieties of both molecules of 20-deoxy 5’-trityithymidine with benzene and toluene are in the C(2’)-endo conformations while molecules A and B of 2’-deoxy 5’-tritylthymidine containing xylene, trimethylbenzene, cyclohexane and water are in the C(3’)-endo and C(1’)-exo conformations, respectively. Both ribose moieties of 5’-TU show C(3’)-endo puckering. The conformation about the C(4’)—C(5’) bond for all the 2’-deoxy 5’-tritylthymidine structures is $g^+$, contrasting with the $g^-$ for the 5’-TU structure. Benzene and toluene molecules stack between TT base pairs, while xylene, trimethylbenzene and cyclohexane are oriented obliquely to the base pairs. 2’-Deoxy 5’-tritylthymidine containing toluene shows a type V C-H…\pi interaction between the methyl group of toluene and the thymine base. Remarkably, the 2’-deoxy 5’-tritylthymidine-containing xylene, trimethylbenzene, cyclohexane and water structures demonstrate a strong type I O-H…\pi interaction between the ribose O(3’) and the thymine base seen only in 1.25% of the structures. Molecular packing and hydrogen bonding are discussed.

Noise characteristics of electro-absorptive logic device utilizing asymmetric Fabry-Perot etalon structure in high optical power

We studied the noise characteristics of a symmetric self electro-optic effect device (S-SEED) with an impedance-matched asymmetric Fabry-Perot cavity and extremely shallow quantum wells (AE-SEED). The device characteristics such as the signal tolerance and the bitrate in cascaded optical logic gates are determined by the parameters including the reflection change Δ R and the contrast ratio CR during the high power operation. The noise characteristics of an AE-SEED were compared with that of a conventional E-SEED. When there was no power fluctuation in the reading beam, the conventional E-SEED showed a higher bitrate. However, when fluctuation was introduced, as is inevitable in the real world, the performance of the AE-SEED as a logic gate was much better and showed higher a bitrate than the E-SEED.

S-SEED operation with noise: bit error rate analysis

The static and dynamic noise characteristics of a symmetric self electrooptic effect device (S-SEED) are studied. The mechanism for bit errors is identified, and the bit error rate related to the decision threshold, input contrast ratio, nominal input power, and bit activation period. Dynamic simulations use practical S-SEED parameters for the diode capacitance, bias voltage, and responsivity values to show specific examples; however, the bit error rate curves provide general results since these parameters are allowed to be variables. The probability of abnormal operation, when there is distortion or an error in the output, is also derived and can be directly related to the bit error rate. In a practical S-SEED switching environment example, we show that the noise effect results in a probability of abnormal operation to be less than 10/sup -50/ provided the bit activation period exceeds the switching time by 0.5%.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Optical energy considerations for diode-clamped smart pixel optical receivers

We calculate the switching time as a function of optical energy for a single-stage smart pixel receiver. We find that operating the receivers dynamically using modelocked pulses is the most energy-efficient method of operation. We also show that the receivers no longer operate with constant optical input energy, like the symmetric self electro-optic effect device (S-SEED), but rather the product of the required optical energy and the switching time is constant, as a result of the introduction of electrical voltage gain.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Light power dependence of electro-optical transmission in InGaAs/AlGaAs multiple quantum wells

Dual wavelength electro-optical bistability in resistor-biased and symmetric self electro-optic effect devices and their input power dependence are investigated using InGaAs/AlGaAs multiple quantum wells. A new principle to drive the devices using two widely separated wavelengths for signal and control light is proposed and demonstrated to obtain optical transmission bistability between two stable states of the signal light at λ1 by varying the control beam power at λ2. Detailed experimental results of signal power dependencies of the optical responses are presented and rigorously explained based on the voltage switching mechanism of the signal diode by the second control light using a simple load line analysis. Advantages of this new device configuration are discussed taking full merit of the transparency of the GaAs substrate to the signal wavelength.