Vertical Tangent Research Articles

Global bifurcation for monotone fronts of elliptic equations

We present two results on global continuation of monotone front-type solutions to elliptic PDEs posed on infinite cylinders. This is done under quite general assumptions, and in particular applies even to fully nonlinear equations as well as quasilinear problems with transmission boundary conditions. Our approach is rooted in the analytic global bifurcation theory of Dancer (1973) and Buffoni–Toland (2003), but extending it to unbounded domains requires contending with new potential limiting behavior relating to loss of compactness. We obtain an exhaustive set of alternatives for the global behavior of the solution curve that is sharp, with each possibility having a direct analogue in the bifurcation theory of second-order ODEs.As a major application of the general theory, we construct global families of internal hydrodynamic bores. These are traveling front solutions of the full two-phase Euler equation in two dimensions. The fluids are confined to a channel that is bounded above and below by rigid walls, with incompressible and irrotational flow in each layer. Small-amplitude fronts for this system have been obtained by several authors. We give the first large-amplitude result in the form of continuous curves of elevation and depression bores. Following the elevation curve to its extreme, we find waves whose interfaces either overturn (develop a vertical tangent) or become exceptionally singular in that the flow in both layers degenerates at a single point on the boundary. For the curve of depression waves, we prove that either the interface overturns or it comes into contact with the upper wall.

Stability of Nonlinear Vibrations of Elastic Plate and Dynamic Absorber in Random Excitations

In this work, the problem of exploring the stability of vibrations of a hysteresis-type elastic dissipative characteristic plate with a liquid section dynamic absorber under the influence of random excitations is considered. Expressions of mean square deviations of the generalized coordinates are presented, and the expression of the spectral density of the base acceleration is obtained in the form of a wide band. The integral expressions of mean square deviations were calculated, and the conditions for the existence of vertical tangents transferred to the graph of the function representing the mean square values of displacements of plate points were determined. It is shown that these conditions are not fulfilled, and the stability condition is determined depending on the structural parameters of the system.

3D Light Scanner Analysis of the Lateral Collateral Ligaments of the Ankle: An Anatomical Study

Category: Basic Sciences/Biologics; Ankle Introduction/Purpose: Recently an anatomical study described a medial connection between the anterior talofibular ligament inferior fascicle (ATFLif), calcaneofibular ligament (CFL) and posterior talofibular ligament (PTFL). Therefore, there is no research evaluating the lateral ligaments and their relationships through a three-dimensional view achieved by a digital marking of their foot prints. The authors hypothesize that a three-dimensional analysis made only by an accurately digital analysis can clarify the lateral collateral ligaments (LCL) connections and their foot prints relations. This work aims to carry out an anatomical study in cadaveric pieces in which the lateral ankle ligaments were dissected and evaluated by a tridimensional scanner that allows a digitally and three-dimensional assessment of their foot prints and connections. Methods: An anatomical study was performed in 11 fresh-frozen ankle specimens. Each ankle was dissected to achieve the amputation of the calcaneus, talus and fibula, keeping the inserts of the footprints of LCL. After that, each bone piece with the insertion remnants of the LCL was submitted to an evaluation by a scanner machine. Surface scans for these bones were collected using a structured light scanner. Each bone piece was set on a 3D automatic turntable to capture a 360° scan and was scanned in two orientations, with 6 scans per orientation. Finally, all the scans were aligned and final fused models were created using the HP DAVID software package and exported to a wavefront 3D object file. The 3D models were imported into Geomagic Studio to calculate surface area of the LCL foot prints. Results: A total of 11 units of each, talus, fibula and calcaneus were dissected from 11 ankles specimens. The ATFL was identified as a two-fascicled ligament and the fibers connecting fibers between the ATFLif and CFL was present in all specimens. In two talus there was only one insertion of the ATFL. After scanning the bones, the authors observed that the foot prints of ATFLif, CFL and PTFL were contiguous at medial side of the fibula, with an average of the surface area of 486,7 mm2. The foot print of the CFL on the calcaneus was located at the junction of a vertical line tangent to the highest point of the posterior subtalar with a horizontal line extending from the sinus tarsi in all specimens. Conclusion: This study demonstrated by a 3D analysis that the LCL of the ankle share the same foot print at medial side on the fibula and are all connected since the foot prints of the ATFL inf, CFL and PTFL were contiguous. In the case of associated proximal lesions of the ATFLif and CFL, ligaments repair with a single suture may be considered. This can be applied in surgical procedures in patients with lateral ankle instability.fig

Continuous Maps Admitting No Tangent Lines: A Centennial of Besicovitch Functions

One of the most influential examples in analysis is a Weierstrass function from to that is continuous but differentiable at no point. However this map, as well most of the others among the myriad similar examples, still admits vertical tangent lines. The examples of continuous maps that admit no tangent line in any direction are also known; however, all currently existing presentations of such maps are not easily accessible due to their very complicated descriptions and hard-to-follow proofs of their desired properties. The goal of this article is to present in an accessible way two such examples. The first—a coordinate of the classical Peano space-filling curve—is simpler, but admits one-sided vertical tangent lines at some points. The second is a variation of a function from a 1924 paper of Besicovitch, which is continuous but admits no one-sided tangent line in any direction. The proofs of nondifferentiability of these two examples will be facilitated by a simple yet general lemma that also implies nondifferentiability of other similar maps, including those of Takagi and van der Waerden.

PARALLELISM OF DISTRIBUTIONS AND GEODESICS ON F(±a2; ±b2)-STRUCTURE LAGRANGIAN MANIFOLD

This paper deals with the Lagrange vertical structure on the vertical space TV (E) endowed with a non null (1,1) tensor field FV satisfying (Fv2-a2)(Fv2+a2)(Fv2 - b2)(Fv2 + b2) = 0. In this paper, the authors have proved that if an almost product structure P on the tangent space of a 2n-dimensional Lagrange manifold E is defined and the F(±a2; ±b2)-structure on the vertical tangent space TV (E) is given, then it is possible to define the similar structure on the horizontal subspace TH(E) and also on T(E). In the next section, we have proved some theorems and have obtained conditions under which the distribution L and M are r-parallel, r¯ anti half parallel when r = r¯ . The last section is devoted to proving theorems on geodesics on the Lagrange manifold

Spatially Quasi-Periodic Water Waves of Infinite Depth

We formulate the two-dimensional gravity-capillary water wave equations in a spatially quasi-periodic setting and present a numerical study of solutions of the initial value problem. We propose a Fourier pseudo-spectral discretization of the equations of motion in which one-dimensional quasi-periodic functions are represented by two-dimensional periodic functions on a torus. We adopt a conformal mapping formulation and employ a quasi-periodic version of the Hilbert transform to determine the normal velocity of the free surface. Two methods of time-stepping the initial value problem are proposed, an explicit Runge–Kutta (ERK) method and an exponential time-differencing (ETD) scheme. The ETD approach makes use of the small-scale decomposition to eliminate stiffness due to surface tension. We perform a convergence study to compare the accuracy and efficiency of the methods on a traveling wave test problem. We also present an example of a periodic wave profile containing vertical tangent lines that is set in motion with a quasi-periodic velocity potential. As time evolves, each wave peak evolves differently, and only some of them overturn. Beyond water waves, we argue that spatial quasi-periodicity is a natural setting to study the dynamics of linear and nonlinear waves, offering a third option to the usual modeling assumption that solutions either evolve on a periodic domain or decay at infinity.

Ultimate Capacity of Vertical Pile Subjected to Combination of Vertical and Lateral Load

Pile under general condition is subjected to combination of vertical and lateral loads In the analytical approaches to predict the load-displacement responses of a pile under central inclined load, it is assumed that the lateral displacement of the pile head is independent by the vertical load factor of the inclined load. Similarly, while estimating the ultimate resistance it is considered that the vertical load factor of the inclined load does not influence the ultimate lateral resistance of the pile during determination of ultimate load carrying capacity of vertical pile. In the present work, an empirical relation has been developed to predict the ultimate load carrying capacity of vertical piles subjected to combination of both vertical and lateral load in cohesion less soil. Effect of lateral load on vertical load deflection behavior of vertical piles when axial loads are present are discussed through several experimental results obtained from tests on model piles. Ultimate capacity is found to be a continuous function of ultimate lateral load, ultimate vertical load capacity and tangent of angle of resultant load made with vertical axis of pile.

Bed shear stress, surface shape and velocity field near the tips of dam-breaks, tsunami and wave runup: Reply by Peter Nielsen

A reply is offered to the discussion by Baldock, T E, (2018): Bed shear stress, surface shape and velocity field near the tips of dam-breaks, tsunami and wave runup. Discussion, Coastal Eng, Vol 142, pp 77–81, henceforth B18 of the swash tip model by Nielsen, P (2018): Bed shear stress, surface shape and velocity field near the tips of dam-breaks, tsunami and wave runup. Coastal Engineering, Vol 138, pp 126–131, henceforth N18. The point made by B18, that the near uniform velocity profiles predicted by the N18 model close to the contact point are unrealistic, is substantiated via new data with enhanced vertical resolution due to large bed roughness, ks ≈ 6.5cm. This data shows that the tip does not actually have a vertical tangent at the contact point. Hence, the main argument for a vertically uniform velocity at a tip, propagating with constant form, falls away. When the N18 model is applied using a measured, non-bullnose, near-tip surface shape it provides realistic velocity profiles and bed shear stress magnitudes. Thus, N18 provides a step towards an integrated explicit 2DV description of swash tips, though not a definitive description.

03:36 PM Abstract No. 173 Effect of shunt geometric parameters on transjugular intrahepatic portosystemic shunt patency

To assess the effect of shunt geometric parameters on transjugular intrahepatic portosystemic shunt (TIPS) patency and to quantitatively analyzed change of stent configuration. A database of patients who had TIPS performed from November 2011 to June 2015 was generated. The angiographic images during initial TIPS creation were reviewed independently by two observers. Stent-to-inferior venous cava distance (SIVCD), hepatic vein to stent angle (HVSA) and portal vein to stent angle (PVSA) was measured in two orthogonal planes. The patients meet the standard that SIVCD < 15mm, HVSA < 45° and PVSA < 45° were classified into the optimal position (OP) group, others classified into non-optimal position (NOP) group. To quantitatively compare the configuration change of stent-graft, the angle between vertical line and tangent of proximal end of stent (AVTP), distal end (AVTD) and stent bending angle (SBA) were measured in portographic images during TIPS creation and follow-up, respectively. Cox regression analysis was performed to assess these parameters on primary patency of TIPS, hepatic encephalopathy and survival. Among all 495 patients, 327 patients were enrolled in OP group. During the follow-up of mean 698 days, the TIPS dysfunction rate among the patients in OP group was 6.1%, compared with 30.0% among the patients in NOP group (P<0.001). Portogram was conducted in 40 patients during the follow-up period. AVTP was significantly decreased by 5.1±8.7° (P = 0.001) and SBA was significantly increased by 9.2±8.5° (P < 0.001). It suggested that stent had tendency to straighten, but the variation was not correlated with time (P = 0.342). The cumulative TIPS patency rates at 1, 2 and 4 years were 93%, 87% and 73%. A Cox multivariate analysis revealed that optimal position of stent (HR = 4.948; 95% CI: 2.929-8.358; P < 0.001) and splenectomy (HR = 2.410; 95% CI: 1.335-4.350; P = 0.004) was associated with TIPS dysfunction. Initial stent position is predictive of TIPS patency, with stent extending to the hepatocaval junction and reducing the angle with vessels. The stent-graft will gradually straighten after TIPS, thereby impacting the shunt patency.Tabled 1Comparison of Geometric Parameters and Prognosis Between the OP and NOP GroupsOP GroupNOP GroupP ValuePatients (n)327168SIVCD (mm)8.8±3.411.6±4.5mm0.002HVSA-a (°)30.2±8.635.1±11.60.042HVSA-l (°)17.7±7.217.1±7.50.203PVSA-a (°)30.2±8.734.9±10.00.014PVSA-l (°)17.0±8.427.7±18.60.001Diameter of stent-graft 10 mm297 (90.8%)147 (87.5%)0.265 8mm36 (9.2%)21 (12.5%)0.134Length of stent-graft 6cm279 (85.3%)136 (81.0%)0.098 8cm48 (14.7%)32 (19.0%)0.148Target portion of PV Right branch248 (75.8%)90 (53.6%)0.021 Left branch42 (12.9%)67 (39.9%)0.002 Bifurcation36 (11.0%)10 (5.9%)0.017 Trunk1 (0.3%)1 (0.6%)0.428PSG decrease (mm Hg)15.7±5.216.5±6.20.266TIPS dysfunction20 (6.1%)47 (30.0%)<0.001 Variceal rebleeding17 (5.2%)36 (21.4%) Ascites recurrence1 (0.3%)0 (0) No symptom2 (0.6%)11 (8.6%)Hepatic encephalopathy103 (31.5%)48 (28.6%)0.403 One time41 (12.5%)20 (11.9%) Two times and above62 (19.0%)28 (16.7%)Mortality43 (13.1%)30 (17.9%)0.118 GI bleeding4 (1.2%)5 (3.0%) Other reason39 (11.9%)25 (14.9%) Open table in a new tab

Removing ring artifacts in cone-beam CT via TV-Stokes and unidirectional total variation model.

It often happens that cone-beam corrupted tomography (CBCT) images have some ring artifacts due to the inconsistent response of detector pixels. Removing ring artifacts in CBCT images without impairing the image quality is critical. The purpose of this study is to implement a whole new method of removing ring artifacts in CBCT images based on TV-Stokes denoising equationand unidirectional total variation (UTV). This method is based on the polar coordinates, where the ring artifacts are shown as horizontal parallel stripes. To begin with, we design a UTV model with the constraint of zero divergence condition to only smooth vertical tangent vectors, and keep horizontal tangent vectors unchanged. In turn, the corresponding smooth normal vectors can be obtained. Next, in order to reconstruct the clean image that fits the obtained normal vectors, the UTV model about the potential clean image is added to the original TV-Stokes denoising equation. Our method was applied to simulated data and real corrupted data to verify its performance. High-quality corrected images without ring artifacts were obtained. In the simulated experiments, our method is not only able to obtain the most complete corrected images, but also capable of acquiring the best quantitative assessments among several different methods. In the experiments of real data, the proposed method was effective in removing ring artifacts and in preserving the original details. Comparative results on several experiments illustrated that our algorithm corrects ring artifacts effectively and outperforms the two compared methods on objective indices and subjective image quality. Benefiting from the TV-Stokes equationand the UTV model, the proposed scheme can be effectively applied to remove ring artifacts in CBCT images, simultaneously preserving the original image details and structure well. Furthermore, this new algorithm can be applied directly on reconstructed images, thus eliminating the requirement for additional imaging data.

The optimal pathway for S1 sacroiliac screwing: 3D imaging study and clinical application

Objective To determine the optimal pathway for S1 sacroiliac screwing based on the largest safe zone in Chinese population using 3D imaging models and observe its clinical application in patients with sacroiliac joint fracture or dislocation. Methods Pelvic CT scans of 54 Chinese adults were obtained to create reconstruction 3D models of the right hemi-pelvis. After the models were transparentized and rotated to the axial view of the S1 pedicle, the angle was slightly adjusted to maximize the translucent safe zone. Next, simulative insertion into the center of this zone was conducted with one virtual screw, as large as possible and tangent to the boundary. Measured were the diameter and length of the screw, the vertical distances from the entry point to the horizontal tangent line of the greater sciatic notch and to the vertical tangent line of the posterior superior iliac spine, and from the exit point to the S1 superior endplate and to the anterior cortex. The differences between males and females were analyzed. The above parameters of the optimal pathway for S1 sacroiliac screwing were used in insertion of 16 screws in the 12 patients with sacroiliac joint fracture or dislocation from January 2014 to January 2016 at Department of Orthopaedics and Traumatology, Nanfang Hospital. They were 8 males and 4 females, from 16 to 47 years of age (average, 34 years). According to the Tile classification, 6 cases belonged to Type Ⅱ and 6 ones to Type Ⅲ. The efficacy of S1 sacroiliac screwing was observed. Results In all the pelvic 3D imaging models, an oval translucent area for safe screw insertion could be easily identified from the S1 pedicle axial view. The maximum diameter and length of the optimal intraosseous pathway were 13.66±2.04 mm and 77.66±4.25 mm; the vertical distances from the entry point to the horizontal tangent line of the greater sciatic notch and to the vertical tangent line of the posterior superior iliac spine were 32.77±4.55 mm and 49.57±5.24 mm; the vertical distances from the exit point to the S1 superior endplate and to the anterior cortex were 9.30±1.54 mm and 15.85±2.12 mm. The differences were of statistical significance between males and females regarding the maximum diameter, the distance from the entry point to the vertical axis, and the distance from the exit point to the anterior cortex (P<0.05). All the 16 screws were safely implanted in the 12 patients. Conclusion The optimal screw pathway can be easily identified and its parameters can be measured in pelvic 3D imaging models using computer virtual technology. The clinical application has proved that the parameters can serve as a theoretical basis for safe placement of S1 sacroiliac screws. Key words: Sacroiliac joint; Bone nails; Fracture fixation, internal; Computer virtual technology; Anatomy

φ-Recurrent Sasakian Finsler Structures

In this paper, φ-recurrent Sasakian Finsler structures on horizontal and vertical tangent bundle diffusions and their various geometric properties are studied.

A threshold value of 3.5 mm of passive anterior tibial subluxation on MRI is highly specific for complete ACL tears

To identify and quantify passive anterior tibial subluxation on MRI using a standardized measurement protocol and determine the diagnostic threshold of subluxation for complete anterior cruciate ligament tears. A retrospective case-control study was performed. Patients who underwent surgery for a complete isolated ACL tear between 2009 and 2015 were matched for age and gender to controls with an intact ligament on knee MRI. All subjects underwent 1.5T closed field MR imaging with the same protocol. Measurements were performed on axial sequences to evaluate translation of the medial and lateral condyles compared to the tibial plateau. Each compartment was measured between the vertical tangent to the posterior femoral condyles and the most posterior part of the tibial plateau. The main criterion was global passive subluxation measurements on MRI, corresponding to mean medial and lateral compartment subluxation. The reproducibility and diagnostic value of passive subluxation were calculated. Sixty (30/30) subjects were included, mean age 27.1 ± 1.7years, 20 women and 40 men. Patients had a significantly higher global passive subluxation than controls (3.3 ± 0.6mm vs 0.6 ± 0.2mm, respectively p < 0.00001). Reproducibility was excellent and the diagnostic value of passive subluxation for a complete ACL tear was fair. A passive subluxation threshold of 3.5mm had a sensitivity of 55.2%, a specificity of 100% and 77.6% of well-classified subjects. The calculated cutoff value for global passive subluxation to identify patients with a complete ACL tear was 3.5mm, with excellent specificity and a high positive likelihood ratio. Suboptimal clinical results following ACL reconstruction could be partially due to failure to restore an anatomical femorotibial relationship. III.

Uncovered Medial Meniscus Sign on Knee MRI: Evidence of Lost Brake Stop Mechanism of the Posterior Horn Medial Meniscus.

The objective of our study was to evaluate the association between anterior tibial translation and injuries on the posterior horn medial meniscus (PHMM) and the integrity of the brake stop mechanism of the PHMM in the anterior cruciate ligament (ACL)-deficient knee. This retrospective study included 85 consecutive patients with an arthroscopically confirmed complete ACL tear. Anterior tibial translation was quantitatively measured using sagittal MRI at the midpoint of the lateral femoral condyle. The "uncovered medial meniscus" sign was considered positive if a vertical line tangent to the posteriormost margin of the medial tibial plateau intersected the PHMM at the midpoint of the medial femoral condyle on sagittal MRI. Concomitant injuries on the structures of the posteromedial and posterolateral corners of the knee, including PHMM tear and meniscal ramp lesion, were recorded. Stratified subgroup analysis and multivariable regression analysis were performed to identify factors associated with anterior tibial translation. The uncovered medial meniscus sign was positive in 21.2% (18/85) of patients and was significantly associated with anterior tibial translation. In the stratified subgroup analysis and multivariable regression analysis, positive uncovered medial meniscus sign consistently showed a significant association with anterior tibial translation and generated an additional 2.8 mm of anterior tibial translation. Other injuries, including PHMM tear and meniscal ramp lesion, were not associated with anterior tibial translation. The uncovered medial meniscus sign showed a statistically significant correlation with anterior tibial translation and could be a useful marker for the lost brake stop mechanism of PHMM in the ACL-deficient knee.

On Generalized and Extended Generalized $\phi$-recurrent Sasakian Finsler Structures

In this research, generalized and extended generalized $\phi $-recurrent Sasakian Finsler structures on horizontal and vertical tangent bundles and their various geometric properties are studied.

Evaluation of Andrews' Analysis as a Predictor of Ideal Sagittal Maxillary Positioning in Orthognathic Surgery

There is no universally accepted method for determining the ideal sagittal position of the maxilla in orthognathic surgery. In "Element II" of "The Six Elements of Orofacial Harmony," Andrews used the forehead to define the goal maxillary position. The purpose of this study was to compare how well this analysis correlated with postoperative findings in patients who underwent bimaxillary orthognathic surgery planned using other guidelines. The authors hypothesized that the Andrews analysis would more consistently reflect clinical outcomes than standard angular and linear measurements. This is a retrospective cohort study of patients who had bimaxillary orthognathic surgery and achieved an acceptable esthetic outcome. Patients with no maxillary sagittal movement, obstructive sleep apnea, cleft or craniofacial diagnoses, or who were non-Caucasian were excluded. Treatment plans were developed using photographs, radiographs, and standard cephalometric measurements. The Andrews analysis, measuring the distance from the maxillary incisor to the goal anterior limit line, and standard measurements were applied to end-treatment records. The Andrews analysis was statistically compared with standard methods. There were 493 patients who had orthognathic surgery from 2007 through 2014, and 60 (62% women; mean age, 22.1±6.8yr) met the criteria for inclusion in this study. The mean Andrews distances were -4.8±2.9mm for women and -8.6±4.6mm for men preoperatively and -0.6±2.1mm for women and -1.9±3.4mm for men postoperatively. For women, the Andrews analysis was closer to the goal value (0mm) postoperatively than any standard measurement (P<.001). For men, the linear distance from the A point to a vertical line tangent to the nasion from the McNamara analysis performed best (P<.001), followed by the Andrews analysis. The Andrews analysis correlated well with the final esthetic sagittal maxillary position in the present sample, particularly for women, and could be a useful tool for orthognathic surgical planning.

Snap-through of shallow reticulated domes under unilateral displacement control

Snap-through instability of shallow reticulated (lattice) domes subjected to a quasi-static downward displacement or force at a joint is analyzed. For the case of unilateral displacement control, the joint is pushed downward by an indentor until it snaps (jumps) to another equilibrium configuration, and then pushed further until another snap occurs, and so on. Under force control, the magnitude of the force is increased, and a different sequence of snaps (local and then global) is exhibited. Green–Lagrange strain is assumed, as well as engineering strain for the smaller dome. The equilibrium equations are solved numerically using Mathematica. For force control, snaps may occur on the equilibrium path (force versus displacement at the force) at limit (maximum) points and at bifurcation points. For unilateral displacement control, snaps may occur at points where the associated downward force decreases to zero, at turning points (which have a vertical tangent), and at bifurcation points (if the bifurcating path moves downward at the loaded joint).

Atiyah classes and dg-Lie algebroids for matched pairs

For every Lie pair $(L,A)$ of algebroids we construct a dg-manifold structure on the $\mathbb{Z}$-graded manifold $\mathcal M=L[1]\oplus L/A$ such that the inclusion $\iota: A[1] \to \mathcal M$ and the projection $p:\mathcal M\to L[1]$ are morphisms of dg-manifolds. The vertical tangent bundle $T^p\mathcal M$ then inherits a structure of dg-Lie algebroid over $\mathcal M$. When the Lie pair comes from a matched pair of Lie algebroids, we show that the inclusion $\iota$ induces a quasi-isomorphism that sends the Atiyah class of this dg-Lie algebroid to the Atiyah class of the Lie pair. We also show how (Atiyah classes of) Lie pairs and dg-Lie algebroids give rise to (Atiyah classes of) dDG-algebras.

Study of epidermis characters and venation of certain species of Medicago L. in Leguminosae (Fabaceae) family in Iraq

The anatomical study of the epidermis leaflet for seven species and variety wild belonging to the genus Medicago L. species are: M. constricta Dur., M. coronata L., M. intertexta L., M. intertexta.var. ciliaris L., M. laciniata L., M. lupulina L., M. minima L. and M. sativa L. were studied, The search included epidermis characters and stomatal complexes addition to venation system in leaflets. It is revealed through the study, epidermis leaflet type Amphistomatic (the stomata spread on the upper and lower surface) as well as the presence of three types of stomatal complexes namely: Anisocytic (the guard cells surrounded by three unequal cell size), Anomocytic (not differential from subsidiary cells in epidermis) and Anomotetracytic (four cells surrounding the guard cell is irregular and varied). The results showed variations in dimensions and forms stomata and nature vertical walls and tangent to the upper and lower surfaces. The venation system distinguish by all the species studied as one of the type Simple craspedodromous in this system all secondary veins and branching ends at the margin of the blade leaflets, this found in all species. But there are clear differences in the Areoles shape and type of vienlets and at the ends, where these characters taxonomic and diagnostic value of the task in the species under study.

Study of epidermis characters and venation of certain species of Medicago L. in Leguminosae (Fabaceae) family in Iraq

The anatomical study of the epidermis leaflet for seven species and variety wild belonging to the genus Medicago L. species are: M. constricta Dur., M. coronata L., M. intertexta L., M. intertexta.var. ciliaris L., M. laciniata L., M. lupulina L., M. minima L. and M. sativa L. were studied, The search included epidermis characters and stomatal complexes addition to venation system in leaflets. It is revealed through the study, epidermis leaflet type Amphistomatic (the stomata spread on the upper and lower surface) as well as the presence of three types of stomatal complexes namely: Anisocytic (the guard cells surrounded by three unequal cell size), Anomocytic (not differential from subsidiary cells in epidermis) and Anomotetracytic (four cells surrounding the guard cell is irregular and varied). The results showed variations in dimensions and forms stomata and nature vertical walls and tangent to the upper and lower surfaces. The venation system distinguish by all the species studied as one of the type Simple craspedodromous in this system all secondary veins and branching ends at the margin of the blade leaflets, this found in all species. But there are clear differences in the Areoles shape and type of vienlets and at the ends, where these characters taxonomic and diagnostic value of the task in the species under study.