Improving Video Watermarking through Galois Field GF(2^(4)) Multiplication Tables with Diverse Irreducible Polynomials and Adaptive Techniques

Video watermarking plays a crucial role in protecting intellectual property rights and ensuring content authenticity.This study delves into the integration of Galois Field(GF)multiplication tables,especially GF(2^(4)),and their interaction with distinct irreducible polynomials.The primary aim is to enhance watermarking techniques for achieving imperceptibility,robustness,and efficient execution time.The research employs scene selection and adaptive thresholding techniques to streamline the watermarking process.Scene selection is used strategically to embed watermarks in the most vital frames of the video,while adaptive thresholding methods ensure that the watermarking process adheres to imperceptibility criteria,maintaining the video's visual quality.Concurrently,careful consideration is given to execution time,crucial in real-world scenarios,to balance efficiency and efficacy.The Peak Signal-to-Noise Ratio(PSNR)serves as a pivotal metric to gauge the watermark's imperceptibility and video quality.The study explores various irreducible polynomials,navigating the trade-offs between computational efficiency and watermark imperceptibility.In parallel,the study pays careful attention to the execution time,a paramount consideration in real-world scenarios,to strike a balance between efficiency and efficacy.This comprehensive analysis provides valuable insights into the interplay of GF multiplication tables,diverse irreducible polynomials,scene selection,adaptive thresholding,imperceptibility,and execution time.The evaluation of the proposed algorithm's robustness was conducted using PSNR and NC metrics,and it was subjected to assessment under the impact of five distinct attack scenarios.These findings contribute to the development of watermarking strategies that balance imperceptibility,robustness,and processing efficiency,enhancing the field's practicality and effectiveness.

Observations on Irreducible Sets and Sober Spaces

This paper, using the monads theory in the topological space, gives a new characterization of irreducible sets in the nonstandard enlarged models. Further, the discretization expression of Sober topological spaces is presented.

REMARKS ON BRUALDI'S THEOREM FOR SPECTRUM INCLUSION REGION OF AN IRREDUCIBLE MATRIX

This paper obtains a necessary and sufficient condition for an irreducible complex matrix whose comparison matrix is a singular M-matrix to be singular. This is used to establish a necessary and sufficient condition for a boundary point of Brualdi’s inclusion region of the eigenvalues of an irreducible complex matrix to be an eigenvalue.

Strongly Cyclic and Strongly Irreducible Decomposition

Let W be an injective unilateral weighted shift, and let W(n) be the orthogonal direct sum of n copies of W. In this paper, we prove that, if the commutant of W is strictly cyclic, then W(n) has a unique (SI) decomposition with respect to similarity for every natural number n.

A new model for predicting irreducible water saturation in tight gas reservoirs

The irreducible water saturation(Swir) is a significant parameter for relative permeability prediction and initial hydrocarbon reserves estimation.However,the complex pore structures of the tight rocks and multiple factors of the formation conditions make the parameter difficult to be accurately predicted by the conventional methods in tight gas reservoirs.In this study,a new model was derived to calculate Swir based on the capillary model and the fractal theory.The model incorporated different types of immobile water and considered the stress effect.The dead or stationary water(DSW) was considered in this model,which described the phenomena of water trapped in the dead-end pores due to detour flow and complex pore structures.The water film,stress effect and formation temperature were also considered in the proposed model.The results calculated by the proposed model are in a good agreement with the experimental data.This proves that for tight sandstone gas reservoirs the Swir calculated from the new model is more accurate.The irreducible water saturation calculated from the new model reveals that Swir is controlled by the critical capillary radius,DSW coefficient,effective stress and formation temperature.

An irreducible polynomial functional basis of two-dimensional Eshelby tensors

The two-dimensional (2D) Eshelby tensors are discussed. Based upon the complex variable method, an integrity basis of ten isotropic invariants of the 2D Eshelby tensors is obtained. Since an integrity basis is always a polynomial functional basis, these ten isotropic invariants are further proven to form an irreducible polynomial functional basis of the 2D Eshelby tensors.

ACCURATE COMPUTATION FOR IRREDUCIBLE NONSINGULAR M-MATRIX

Computing the eigenvalue of smallest modulus and its corresponding eigneveclor of an irreducible nonsingular M-matrix A is considered, It is shown that if the entries of A are known with high relative accuracy, its eigenvalue of smallest modulus and each component of the corresponding eigenvector will be determined to much higher accuracy than the standard perturbation theory suggests. An algorithm is presented to compute them with a small componentwise backward error, which is consistent with the perturbation results.

LOWER BOUND NUMBER OF IRREDUCIBLE GRAPHS ON SURMCES

Irreducible graphs are those graphs which can hot be embedded on some surface but deleting ally edge of this graph, the resultant graph can be embedded on the surface. This paper characterizes one kind of irreducible graphs and it is shown that the lower bound number of irreducible graphs for the surface of genus n is 2 × 16n.

Application of Angular Momentum Theory to Constructing Basis Functions of Irreducible Representations of Icosahedral Group

A new method based on angular momentum theory was proposed to construct the basis functions of the irreducible representations(IRs) of point groups. The transformation coefficients, i. e., coefficients S, are the components of the eigenvectors of some Hermitian matrices, and can be made as real numbers for all pure rotation point groups. The general formula for coefficient S was deduced, and applied to constructing the basis functions of single-valued irreducible representations of icosahedral group from the spherical harmonics with angular momentum j≤7.

ORIENTATION DISTRIBUTION FUNCTIONS FOR MICROSTRUCTURES OF HETEROGENEOUS MATERIALS (Ⅱ)──CRYSTAL DISTRIBUTION FUNCTIONS AND IRREDUCIBLE TENSORS RESTRICTED BY VARIOUS MATERIAL SYMMETRIES

The explicit representations for tensorial Fourier expansion of 3_D crystal orientation distribution functions (CODFs) are established. In comparison with that the coefficients in the mth term of the Fourier expansion of a 3_D ODF make up just a single irreducible mth_order tensor, the coefficients in the mth term of the Fourier expansion of a 3_D CODF constitute generally so many as 2m+1 irreducible mth_order tensors. Therefore, the restricted forms of tensorial Fourier expansions of 3_D CODFs imposed by various micro_ and macro_scopic symmetries are further established, and it is shown that in most cases of symmetry the restricted forms of tensorial Fourier expansions of 3_D CODFs contain remarkably reduced numbers of mth_order irreducible tensors than the number 2m+1 . These results are based on the restricted forms of irreducible tensors imposed by various point_group symmetries, which are also thoroughly investigated in the present part in both 2_ and 3_D spaces.

ORIENTATION DISTRIBUTION FUNCTIONS FOR MICROSTRUCTURES OF HETEROGENEOUS MATERIALS (Ⅰ) ──DIRECTIONAL DISTRIBUTION FUNCTIONS AND IRREDUCIBLE TENSORS

In this two_part paper, a thorough investigation is made on Fourier expansions with irreducible tensorial coefficients for orientation distribution functions (ODFs) and crystal orientation distribution functions (CODFs), which are scalar functions defined on the unit sphere and the rotation group, respectively. Recently it has been becoming clearer and clearer that concepts of ODF and CODF play a dominant role in various micromechanically_based approaches to mechanical and physical properties of heterogeneous materials. The theory of group representations shows that a square integrable ODF can be expanded as an absolutely convergent Fourier series of spherical harmonics and these spherical harmonics can further be expressed in terms of irreducible tensors. The fundamental importance of such irreducible tensorial coefficients is that they characterize the macroscopic or overall effect of the orientation distribution of the size, shape, phase, position of the material constitutions and defects. In Part (Ⅰ), the investigation about the irreducible tensorial Fourier expansions of ODFs defined on the N_dimensional (N_D) unit sphere is carried out. Attention is particularly paid to constructing simple expressions for 2_ and 3_D irreducible tensors of any orders in accordance with the convenience of arriving at their restricted forms imposed by various point_group (the synonym of subgroup of the full orthogonal group) symmetries. In the continued work -Part (Ⅱ), the explicit expression for the irreducible tensorial expansions of CODFs is established. The restricted forms of irreducible tensors and irreducible tensorial Fourier expansions of ODFs and CODFs imposed by various point_group symmetries are derived.

Irreducible Traumatic Anterior Shoulder Dislocation Secondary to Both Subscapularis and Long Head of Biceps Tendon Interposition—A Case Report

An Irreducible dislocation of the shoulder is an uncommon event. When it does occur, blocks to reduction can include bone, labrum, rotator cuff musculature or tendon. Patients older than 40 at the time of initial dislocation are at increased risk of sustaining a concomitant rotator cuff tear. We present a case of an irreducible anterior shoulder dislocation due to interposition of both subscapularis tendon and a posteriorly dislocated long head of biceps. Both Computed Tomography (CT) and magnetic resonance imaging (MRI), along with intraoperative findings are discussed. Conclusion: We would advocate maintaining a low threshold for MR imaging post shoulder dislocation in the older population, when there is radiological or clinical concern regarding the integrity of the rotator cuff, and also to evaluate whether a concentric reduction of the shoulder joint has been achieved.

Irreducible Representations of Algebraic Group <i>SL</i>(6,<i>K</i>) in char<i>K</i>=3

For each irreducible module Xi Nanhua defined an element which generated this module. We use this element to construct a certain basis for and then compute dim , determine its formal characters in this paper. In order to obtain faster speed we modify the algorithm to compute the irreducible characters.

Irreducible Polynomials in &#918;[x] That Are Reducible Modulo All Primes

The polynomial x4+1 is irreducible in &#918;[x] but is locally reducible, that is, it factors modulo p for all primes p. In this paper we investigate this phenomenon and prove that for any composite natural number N there are monic irreducible polynomials in &#918;[x] which are reducible modulo every prime.

Search for Monic Irreducible Polynomials with Decimal Equivalents of Polynomials over Galois Field <I>GF</I>(<I>p^q</I>)

Substitution boxes or S-boxes play a significant role in encryption and de-cryption of bit level plaintext and cipher-text respectively. Irreducible Poly-nomials (IPs) have been used to construct 4-bit or 8-bit substitution boxes in many cryptographic block ciphers. In Advance Encryption Standard, the ele-ments of 8-bit S-box have been obtained from the Multiplicative Inverse (MI) of elemental polynomials (EPs) of the 1st IP over Galois field GF(28) by adding an additive element. In this paper, a mathematical method and the algorithm of the said method with the discussion of the execution time of the algorithm, to obtain monic IPs over Galois field GF(pq) have been illustrated with example. The method is very similar to polynomial multiplication of two polynomials over Galois field GF(pq) but has a difference in execution. The decimal equivalents of polynomials have been used to identify Basic Polynomials (BPs), EPs, IPs and Reducible polynomials (RPs). The monic RPs have been determined by this method and have been cancelled out to produce monic IPs. The non-monic IPs have been obtained with multiplication of α where?α∈ GF(pq)?and assume values from 2 to (p &#8722;1) to monic IPs.

Generalized Irreducible α-Matrices and Its Applications

The class of generalized α-matrices is presented by Cvetkovi?, L. (2006), and proved to be a subclass of H-matrices. In this paper, we present a new class of matrices-generalized irreducible α-matrices, and prove that a generalized irreducible α-matrix is an H-matrix. Furthermore, using the generalized arithmetic-geometric mean inequality, we obtain two new classes of H-matrices. As applications of the obtained results, three regions including all the eigenvalues of a matrix are given.

Intracortical screw insertion plus limited open reduction in treating type 31A3 irreducible intertrochanteric fractures in the elderly

BACKGROUND In most elderly patients with intertrochanteric fractures,satisfactory fracture reduction can be achieved by closed reduction using a traction table.However,intertrochanteric fractures cannot achieve satisfactory reduction in a few patients,which is called irreducible intertrochanteric fractures.Especially for type 31A3 irreducible intertrochanteric fractures,limited open reduction of the broken end with different intraoperative reduction methods is required to achieve satisfactory reduction and fixation.AIM To discuss clinical efficacy of intracortical screw insertion plus limited open reduction in type 31A3 irreducible intertrochanteric fractures in the elderly.METHODS A retrospective analysis was performed on 23 elderly patients with type 31A3 irreducible intertrochanteric fractures(12 males and 11 females,aged 65-89-yearsold)who received treatment at the orthopedics department.After type 31A3 irreducible intertrochanteric fractures were confirmed by intraoperative C-arm,all of these cases received intracortical screw insertion plus limited open reduction in the broken end with intramedullary screw internal fixation.The basic information of surgery,reduction effects,and functional recovery scores of the hip joint were assessed.RESULTS All patients were followed up for 13.8 mo on average.The operation time was 53.8±13.6 min(40-95 min).The intraoperative blood loss was 218.5±28.6 mL(170-320 mL).The average number of intraoperative X-rays was 22.8±4.6(18-33).The average time to fracture union was 4.8±0.7 mo.The reduction effect was assessed using Kim’s fracture reduction evaluation.Twenty cases achieved grade I fracture reduction and three cases grade II fracture reduction.All of them achieved excellent or good fracture reduction.Upon the last follow-up,the functional recovery scores score was 83.6±9.8,which was not significantly different from the functional recovery scores score(84.8±10.7)before the fracture(t=0.397,P=0.694).CONCLUSION With careful preoperative preparation,intracortical screw insertion plus limited open reduction contributed to high-quality fracture reduction and fixation.Good clinical outcomes were achieved without increasing operation time and intraoperative blood loss.

Irreducible Posterior Hip Dislocation with Associated Isolated Comminuted Greater Trochanter Avulsion Fracture Treated with Universal Locking Trochanteric Stabilization Plate: A Rare Combination of Hip Injury

Posterior hip dislocation with greater trochanter fracture is an uncommon injury pattern in the acute trauma patient. Frequently associated injury includes a combination of hip dislocation with posterior wall of acetabulum, head of femur fracture, intertrochanteric fracture and even the most severe type of combined acetabular fracture. We report a 42-year-old man post traumatic bilateral hip injuries with irreducible posterior hip dislocation and associated isolated greater trochanteric fracture successfully managed with open reduction and fixation of greater trochanter with universal locking trochanteric stabilization plate.

Generalized Nonlinear Irreducible Auto-Correlation and Its Applications in Nonlinear Prediction Models Identification

There is still an obstacle to prevent neural network from wider and more effective applications, i.e., the lack of effective theories of models identification. Based on information theory and its generalization, this paper introduces a universal method to achieve nonlinear models identification. Two key quantities, which are called nonlinear irreducible auto-correlation (NIAC) and generalized nonlinear irreducible auto-correlation (GNIAC), are defined and discussed. NIAC and GNIAC correspond with intrinstic irreducible auto-(dependency) (IAD) and generalized irreducible auto-(dependency) (GIAD) of time series respectively. By investigating the evolving trend of NIAC and GNIAC, the optimal auto-regressive order of nonlinear auto-regressive models could be determined naturally. Subsequently, an efficient algorithm computing NIAC and GNIAC is discussed. Experiments on simulating data sets and typical nonlinear prediction models indicate remarkable correlation between optimal auto-regressive order and the highest order that NIAC-GNIAC have a remarkable non-zero value, therefore demonstrate the validity of the proposal in this paper.

Applying the Parameter“Irreducible Concentration”in Modelling of Stormwater Treatment Facilities

To design stormwater treatment facilities (STFs), we recommend the use of a model that should include the calculation of runoff quality, to be based on a detailed land use specification, include site-specific design parameters, calculated outflow concentrations and loads of specified pollutants of relevance for the receiving water. This study compiles minimum outflow concentrations from stormwater databases of different types of STFs (e.g. swales, wet ponds, wetlands, biofilters and underground retention basins with filters). These data are used for the suggested values of specific “irreducible concentrations” (Cirr). Suggested Cirr for phosphorus (P) varies from 20-82 μg/L depending on facility type, for copper (Cu) 1.1-3.7 g/L, for zinc (Zn) 2.0-17 μg/L and for total suspended solids (TSS) 2,900-5,700 μg/L. Corresponding data for 70 substances are compiled in the StormTac database and employed in the model StormTac Web. Cirr has significant impact regarding the choice of facility type and its calculated dimensions. This design parameter and the calculated outflow concentrations can be used to investigate the need for combined serial facilities or complemented design elements with more planted vegetation or installed filters. Such elements can be required to decrease Cirr and thereby reach project specific limit outflow concentrations and loads.