18 research outputs found

    μ•”ν˜Έν™”λœ μƒνƒœμ—μ„œμ˜ κΈ°κ³„ν•™μŠ΅κ³Ό λ™ν˜• 비ꡐ연산

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    ν•™μœ„λ…Όλ¬Έ (박사) -- μ„œμšΈλŒ€ν•™κ΅ λŒ€ν•™μ› : μžμ—°κ³Όν•™λŒ€ν•™ μˆ˜λ¦¬κ³Όν•™λΆ€, 2021. 2. μ²œμ •ν¬.κΈ°κ³„ν•™μŠ΅μ€ 졜근 λΆ„μ•ΌλΆˆλ¬Έ 빅데이터 λΆ„μ„μ˜ κ°€μž₯ 보편적인 방법 쀑 ν•˜λ‚˜λ‘œ μΈμ‹λ˜κ³  μžˆμ§€λ§Œ, κΈ°μ—…μ΄λ‚˜ 기관듀이 μ‹€μ œ 데이 터에 ν™œμš©ν•˜κΈ°μ—λŠ” 데이터 ν”„λΌμ΄λ²„μ‹œ λ¬Έμ œκ°€ 큰 우렀둜 λ‚¨μ•„μžˆλ‹€. 데이터 ν”„λΌμ΄λ²„μ‹œ 보호λ₯Ό μœ„ν•΄ μ—¬λŸ¬ λΉ„μ•”ν˜Έν•™μ  방법둠듀이 κ·Έ λ™μ•ˆ ν™œμš©λ˜μ–΄ μ™”μ§€λ§Œ, 각 κ³Όμ • μ†μ—μ„œ λΆˆκ°€ν”Όν•˜κ²Œ μΌμ–΄λ‚˜λŠ” 정보 손싀은 κ²°κ΅­ 데이터 μœ μš©μ„±μ„ ν˜„μ €ν•˜κ²Œ λ–¨μ–΄λœ¨λ¦¬λŠ” 단점을 가지고 μžˆλ‹€. λ™ν˜•μ•”ν˜ΈλŠ” λ³΅ν˜Έν™” 과정없이 μ•”ν˜Έν™”λœ μƒνƒœλ‘œ 연산을 μ§€μ›ν•˜κΈ° λ•Œλ¬Έμ— 데이터 ν”„λ¦¬μ΄λ²„μ‹œμ™€ μœ μš©μ„±μ„ λ™μ‹œμ— λ³΄μ‘΄ν•˜κ³ , κΈ°κ³„ν•™μŠ΅μ— μ μš©ν•˜κΈ°μ— κ°€μž₯ μ ν•©ν•œ μ•”ν˜Έν•™μ  ν”„λ¦¬λ―Έν‹°λΈŒ 쀑 ν•˜λ‚˜λ‘œ 여겨지고 μžˆλ‹€. ν•˜μ§€λ§Œ κ·Έ λ™μ•ˆ 타켓 μ—°μ‚°ν•¨μˆ˜μ˜ κΉŠμ΄κ°€ κΉŠκ±°λ‚˜ λ‹€μˆ˜μ˜ 비닀항식 연산을 ν¬ν•¨ν•˜λŠ” 경우 ꡉμž₯히 연산이 μ˜€λž˜κ±Έλ¦¬λŠ” 문제 λ•Œλ¬Έμ—, μ§€κΈˆκΉŒμ§€ 개발된 λ™ν˜•κΈ°κ³„ν•™μŠ΅ μ•Œκ³ λ¦¬μ¦˜μ˜ λ²”μ£ΌλŠ” ꡉμž₯히 μ œν•œλ˜μ–΄ μžˆμ—ˆλ‹€. λ³Έ λ…Όλ¬Έμ—μ„œλŠ” μ΄λŸ¬ν•œ ν•œκ³„μ μ„ κ·Ήλ³΅ν•˜κΈ° μœ„ν•œ 두가지 방법둠을 μ œμ‹œν•œλ‹€. 첫번째 방법둠은 κΈ°μ‘΄ κΈ°κ³„ν•™μŠ΅ μ•Œκ³ λ¦¬μ¦˜μ˜ 비닀항식 연산을 λ™ν˜•μ•”ν˜Έ μΉœν™”μ μΈ ν˜•νƒœλ‘œ λ³€ν˜•μ‹œν‚€λŠ” 방법이닀. κΈ°κ³„ν•™μŠ΅μ—μ„œ κ°€μž₯ 널리 μ‚¬μš©λ˜λŠ” 기본적인 방법인 λ‘œμ§€μŠ€ν‹± νšŒκ·€λΆ„μ„μ— 이 방법둠을 기반으둜 ν•œ 연ꡬλ₯Ό μ§„ν–‰ν•˜μ˜€κ³ , μ „μž₯μœ μ „μ²΄μ—°κ΄€λΆ„μ„ λ“± μ‹€μ œ 많이 ν™œμš©λ˜λŠ” μ‘μš©λΆ„μ•Όμ— μ μš©ν•˜μ—¬ μ‹€νš¨μ„±μ„ μž…μ¦ν•˜μ˜€λ‹€. λ‘λ²ˆμ§Έ 방법둠은 κΈ°μ‘΄ κΈ°κ³„ν•™μŠ΅ μ•Œκ³ λ¦¬μ¦˜ 자체λ₯Ό λ³€ν˜•ν•˜κΈ°λ³΄λ‹€ κ·Έ λ‚΄λΆ€μ˜ 비닀항식 연산에 λŒ€ν•œ 효율적인 닀항식 근사 방법을 μ°ΎλŠ” 것이닀. λ³Έ λ…Όλ¬Έμ—μ„œλŠ” κ°€μž₯ 널리 μ‚¬μš©λ˜λŠ” 비닀항식 μ—°μ‚°λ“€ 쀑 ν•˜λ‚˜μΈ 비ꡐ연산과 μ΅œλŒ€/μ΅œμ†Œμ—°μ‚°μ„ νƒ€κ²ŸμœΌλ‘œ 연ꡬλ₯Ό μ§„ν–‰ν•˜μ˜€κ³ , ν•©μ„±ν•¨μˆ˜ 근사법을 톡해 이 연산듀에 λŒ€ν•œ 졜적 κ³„μ‚°λŸ‰μ„ κ°€μ§€λŠ” λ™ν˜•μ•”ν˜Έ μ•Œκ³ λ¦¬μ¦˜ κ°œλ°œμ— μ„±κ³΅ν•˜μ˜€λ‹€.As machine learning (ML) has become a universal tool of big data analysis regardless of field, data privacy has emerged one of the most significant issues to be solved for applying ML to real-world applications. Some non-cryptographic methodologies have been applied so far for privacy preservation, but the loss of information is inevitable which leads to significant reduction in data usability. Homomorphic Encryption (HE) has been recognized one of the most appropriate cryptographic primitives for privacy-preserving ML preserving both data privacy and usability, from its beautiful functionality that allows computation over encrypted data without decryption. However, extremely high computational cost of HE computation originated from the large depth of target functions or a number of non-polynomial operations has remained a main bottleneck of applying HE in privacy-preserving ML. In this thesis, we introduce two main methodologies to overcome this limitation. The first one is to modify the existing ML algorithms into HE-friendly form. We instantiate this methodology to logistic regression, which is one of the most popular methods for classification, and show the practicality by applying our method to real-world applications including genome-wide association study (GWAS). The second one is to find efficient polynomial approximation of non-polynomial functions, instead of substituting them with some other HE-friendly operations. Based on composite function approximation methods, we develop complexity-optimal HE algorithms for comparison and min/max functions, which are the most frequently used non-polynomial operations in real-world computation. We also show the practicality of our algorithms by implementing them based on an approximate HE scheme HEaaN: The homomorphic comparison of two encrypted 16-bit integers takes only 1.22 milliseconds in amortized running time, which is 18 faster than the previous best result.1 Introduction 1 1.1 Our Contributions 3 1.2 List of Papers 6 2 Preliminaries 8 2.1 Notations 8 2.2 Mathematical Backgrounds 9 2.2.1 Minimax Polynomial Approximation on Sign Function 9 2.2.2 Gradient Descent with Errors 11 2.3 Homomorphic Encryption 13 2.3.1 Approximate HE Scheme HEaaN 15 2.3.2 Matrix Packing Method in HEaaN 19 2.4 Logistic Regression 20 2.5 Semi-parallel GWAS 21 3 Related Works 24 3.1 HE-based Logistic Regression and Ensemble Methods 24 3.2 Privacy-preserving Genome Analysis 25 3.3 Study on the Comparison Operation 27 3.3.1 Numerical Analysis on the Sign Function 27 3.3.2 Previous Homomorphic Comparison Methods 29 4 Approximate GWAS based on HE 31 4.1 Motivation 32 4.2 Summary of Results 33 4.3 Our Optimization Methodology 34 4.3.1 Our Modi ed semi-parallel GWAS Algorithm 37 4.4 Homomorphic Evaluation of the Modi ed Algorithm 38 4.5 Implementation 46 4.5.1 Dataset Description 46 4.5.2 Experimental Setting and Parameter Selection 46 4.5.3 Experimental Results and Evaluation 47 4.6 Discussion 50 5 Ensemble Logistic Regression based on HE 52 5.1 Motivation 53 5.2 Summary of Results 54 5.3 Our Ensemble Method 55 5.3.1 Ensemble Gradient Descent 56 5.3.2 Statistical Guarantee of Ensemble Gradient Descent 58 5.3.3 Ensemble GD with Errors and its Convergence 59 5.4 Implementation 63 5.4.1 Experimental Settings 63 5.4.2 Experimental Results 65 5.5 Discussion 68 6 Numerical Method for Homomorphic Comparison 70 6.1 Background and Overview 71 6.1.1 High-Level Idea 72 6.1.2 Summary of Results 74 6.2 Iterative Algorithms 76 6.2.1 Inverse 76 6.2.2 Square Root 78 6.3 Approximate min/max Algorithms 79 6.3.1 Min/Max Algorithm for two numbers 80 6.3.2 Min/Max Algorithm for several numbers 83 6.4 Approximate Comparison Algorithms 86 6.4.1 Comparison between two numbers 87 6.4.2 Max Index of several numbers 90 6.5 Asymptotic Optimality of our Methods 93 6.5.1 Min/Max from Minimax Approximation 95 6.5.2 Comparison from Minimax Approximation 96 6.6 Applications of Comparison Algorithms 100 6.6.1 Threshold Counting 100 6.6.2 Top-k Max 102 6.7 Experimental Results 104 6.7.1 Max of two integers 105 6.7.2 Comparison of two integers 106 6.7.3 Other Applications 108 7 Homomorphic Comparison with Optimal Complexity 110 7.1 Background and Overview 111 7.1.1 Our Idea and Technical Overview 113 7.1.2 Summary of Results 117 7.2 Our New Comparison Method 119 7.2.1 Composite Polynomial Approximation on the Sign Function 120 7.2.2 Analysis on the Convergence of f_n^(d) 124 7.2.3 New Comparison Algorithm NewComp 128 7.2.4 Computational Complexity of NewComp 129 7.2.5 Heuristic Methodology of Convergence Acceleration 132 7.3 Application to Min/Max 139 7.4 Additional Properties of f_n and g_n 142 7.4.1 Convergence of delta_0, S and g_{n,tau} 142 7.4.2 Heuristic Properties on g_n 143 7.4.3 Convergence of f_n^(d) in Erroneous Case 144 7.5 Experimental Results 149 7.5.1 Parameter Selection 149 7.5.2 Performance of NewComp and NewCompG 150 7.5.3 Performance of NewMax and NewMaxG 153 8 Conclusion 154 Abstract (in Korean) 174 Acknowledgement (in Korean) 175Docto

    (A)Study on robust and optimal integration of active 4 wheel steering and direct yaw moment control

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    ν•™μœ„λ…Όλ¬Έ(석사) --μ„œμšΈλŒ€ν•™κ΅ λŒ€ν•™μ› :기계항곡곡학뢀,2006.Maste

    뢄사 ν‘œλ©΄μ²˜λ¦¬λœ κ΅μ •μš© λ―Έλ‹ˆμŠ€ν¬λ₯˜ μž„ν”Œλž€νŠΈμ˜ μ‘°μ§ν˜•νƒœν•™μ  연ꡬ

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    Dept. of Dental Science/박사Diabetic peripheral neuropathy (DPN) is the most common complication of diabetic mellitus (DM) that can cause significant morbidity and mortality, and is found in >60% of patients over the course of their disease. DPN is characterized by diffuse or focal damages to peripheral somatic or autonomic nerve fibers which produce dysfunctions of skin, muscles, and visceral organs. The primary risk factor is hyperglycemia which activates multiple biochemical pathways including AGE, polyol, hexosamine, PKC, and PARP pathways to result in cell dysfunctions and death. Generally, DPN can be divided into sensory motor neuropathy and autonomic neuropathy. The latter is further classified as cardiaovascular, gastrointestinal, and genitourinary autonomic neuropathies. Among them, genitourinary autonomic neuropathy has been poorly understood in terms of its cellular and molecular mechanism. The major clinical symptoms of genitourinary autonomic neuropathy are bladder and sexual dysfunctions. Major pelvic ganglion (MPG), located on the lateral surfaces of the prostate gland in rat provides autonomic innervation to the distal colon and urogenital organs including the urinary bladder, the prostate, and the penis. Among autonomic ganglia, MPG is very unique since sympathetic and parasympathetic neurons are colocalized in the same ganglion capsule. Thus, MPG is critical for autonomic reflexes such as micturition and penile erection together with visceral sensory dorsal root ganglion (DRG). To date, it is little known whether DM affects functions of MPG and DRG neurons. Accordingly, I hypothesized that DM causes plastic changes of MPG and DRG neuronal functions by altering expression of certain types of ion channels, which may contribute to autonomic genitourinary dysfunctions. Thus, the purposes of the present study were to test whether excitability of MPG and DRG neurons is altered in an experimantal diabetic rat model, and to define the molecular and cellular mechanisms underlying DM-induced changes in excitability.Experimental DM was induced by injection of streptozocin (STZ, 60 mg/kg, i.p.) into S/D rats (8 week-old). After three days, the levels of blood glucose in control and STZ-injected rats were measured. STZ-injected rats with hyperglycemia (>300 mg/dl) were divided into two groups (STZ and STZ+insluin). Insulin (10 IU) was injected into diabetic rats once per day. After 8 weeks, development of diabetic neuropathy in the STZ group was assessed by measuring motor nerve conduction velocity (MNCV) in sciatic nerves, myelin area, bladder micturition pattern, and intracarvernous pressure (ICP). As results, motor and autonomic neuropathies were found to be developed in the STZ group. H & E staining revealed the enlarged bladder with hypetropied urothelium in the STZ group. DM significantly decreased testosterone level while increased corticosterone, a stress hormone. Interestingly, DM was found to increase serum and tissue oxidative stress as the malondialdehyde level was measured. Under the gramicidine-perforated configuration of the patch-clamp techniques, spike firing was recorded in MPG and capsaicin-sensitive C-fiber DRG neurons (L6-S1). Spike firing frequency was decreased in both sympathetic and parasympathetic MPG neurons, while increased in DRG neurons from the STZ group. However, insulin significantly attenuated the effects of DM on the excitability of MPG and DRG neurons. DM did not alter the passive properties (input impedence, and resting membrane potentials) in MPG and DRG neurons. However, DM significantly increased the duration of afterhyperpolarization (AHP) in MPG neurons, while decreased it in DRG neurons, which might alter the excitability of neurons. Real-time RT-PCR and western blot analyses revealed that expression of T-type 1H Ca2+ channels was down-regulated in MPG, but up-regulated in DRG from the STZ group. Consistent with these molecular data, T-type Ca2+ currents were decreased in sympathetic MPG neurons, but increased in DRG neurons from the STZ group. Furthermore, expression of SK channels determining AHP duration was up-regulated in MPG neurons, while down-regulated in DRG neurons from the STZ group. In in vitro studies, T-type current density was significantly decreased by high glucose and the pro-inflammatory cytokines in MPG neurons. In DRG neurons, however, T-type current density was increased only by high glucose. In addition, H2O2 significantly reduced T-type current density in MPG neurons, while slightly increased it with no statistical significance in DRG neurons. Taken together, experimental DM alters the excitability of autonomic MPG and DRG neurons by differential regulation of expression of T-type Ca2 and SK potassium channels, which might produce autonomic imbalance contributing to the genitourinary dysfunctions. The significances of the present study are as follow; I studied for the first time the effects of DM on functional plasticity of autonomic ganglion neurons innervating the urogenital system. More importantly, I suggest the molecular and cellular mechanisms underlying the DM-induced autonomic plasticity.restrictio

    Anatomical characteristics of the midpalatal suture area for miniscrew implantation using CT image

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    μΉ˜μ˜ν•™κ³Ό/석사[ν•œκΈ€] κ³ μ •μ›μœΌλ‘œμ„œ μž„ν”Œλž€νŠΈλ₯Ό μ‚¬μš©ν•˜κΈ° μœ„ν•΄μ„œλŠ” 무엇보닀 식립후에 μ•ˆμ •μ μ΄μ–΄μ•Ό ν•œλ‹€. 이λ₯Ό μœ„ν•΄μ„œ μ‹λ¦½λΆ€μœ„μ˜ κ³¨μƒνƒœκ°€ μ€‘μš”ν•˜λ‹€. 즉 ν”Όμ§ˆκ³¨μ΄ μΆ©λΆ„ν•œ λ‘κ»˜λ₯Ό κ°€μ§€λŠ” λΆ€μœ„μ— μ‹λ¦½ν•˜μ—¬ μΆ©λΆ„ν•œ 초기 고정을 μ–»λŠ” 것이 μž„ν”Œλž€νŠΈμ˜ μ•ˆμ •μ„±μ„ μœ„ν•΄μ„œ 맀우 μ€‘μš”ν•˜λ‹€. μ΄λŸ¬ν•œ κ΄€μ μ—μ„œ ν•˜μ•…κ³¨μ— λΉ„ν•˜μ—¬ μ „λ°˜μ μœΌλ‘œ 골질이 쒋지 μ•Šκ³  특히 ν˜‘μΈ‘μ—μ„œ ν”Όμ§ˆκ³¨μ˜ λ‘κ»˜κ°€ 얇은 μƒμ•…κ³¨μ—μ„œλŠ” μΆ©λΆ„ν•œ λ‘κ»˜μ˜ ν”Όμ§ˆκ³¨μ„ 가지고 μžˆλŠ” μ •μ€‘κ΅¬κ°œλ΄‰ν•©λΆ€μœ„κ°€ μž„ν”Œλž€νŠΈμ˜ 식립 λΆ€μœ„λ‘œ μ£Όλͺ©μ„ λ°›κ³  μžˆλ‹€. ν•˜μ§€λ§Œ κ΅μ •μš© μž„ν”Œλž€νŠΈ μ‚¬μš©μ— μžˆμ–΄μ„œ μ •μ€‘κ΅¬κ°œλ΄‰ν•©λΆ€μœ„μ˜ 골쑰직의 높이에 λŒ€ν•œ ꡬ체적인 해뢀학적 μžλ£Œκ°€ μ—†μ–΄μ„œ μž„μƒμ— μ μš©μ‹œ 이둠적 κ·Όκ±°κ°€ λΆ€μ‘±ν•œ 것이 μ‚¬μ‹€μ΄μ—ˆλ‹€. 이에 λ³Έ μ—°κ΅¬λŠ” μ „μ‚°ν™”λ‹¨μΈ΅μ΄¬μ˜κ³Ό μž¬κ΅¬μ„± ν”„λ‘œκ·Έλž¨μΈ V works 4.0 (Cybermed Inc., Seoul, Korea)을 μ΄μš©ν•˜μ—¬ 보철용 μž„ν”Œλž€νŠΈ 식립을 μœ„ν•΄ μ΄¬μ˜ν•œ 성인 λ‚¨μž 14λͺ…, 성인 μ—¬μž 14λͺ…μ˜ CT 자료λ₯Ό 톡해 1) 정쀑 μ‹œμƒλ©΄μ—μ„œ μƒμ•…κ³¨μ˜ 길이λ₯Ό κ³„μΈ‘ν•˜κ³  2) 정쀑 ꡬ개 봉합 λΆ€μœ„μ—μ„œ 골쑰직 λ‘κ»˜λ₯Ό μΈ‘μ •ν•˜λ©° 3) κ΅μ •μš© μž„ν”Œλž€νŠΈλ₯Ό μ•ˆμ „ν•˜κ²Œ 식립할 수 μžˆλŠ” ꡬ체적 λΆ€μœ„λ₯Ό μ„€μ •ν•˜κ³ μž ν•˜μ—¬ λ‹€μŒκ³Ό 같은 κ²°κ³Όλ₯Ό μ–»μ—ˆλ‹€. 1. ANSμ—μ„œ PNSκΉŒμ§€μ˜ μƒμ•…κ³¨μ˜ 길이λ₯Ό κ³„μΈ‘ν•œ κ²°κ³Ό λ‚¨μžμ—μ„œλŠ” 평균 51.08mm, μ—¬μžμ—μ„œλŠ” 평균 47.34mmμ˜€λ‹€. 남녀 μ‚¬μ΄μ—λŠ” ν†΅κ³„μ μœΌλ‘œ μœ μ˜ν•œ 차이가 μžˆμ—ˆλ‹€.(p<0.05) 2. μ •μ€‘κ΅¬κ°œλ΄‰ν•©λΆ€μœ„μ˜ 골쑰직 λ‘κ»˜λ₯Ό μΈ‘μ •ν•œ κ²°κ³Ό λ‚¨μžμ˜ ANS-PNS길이의 μ€‘μ μ—μ„œ κ΅¬κ°œν‰λ©΄μ„ 따라 PNSμͺ½μœΌλ‘œ 15mm지점을 μ œμ™Έν•˜κ³ λŠ” 남녀λͺ¨λ‘μ—μ„œ 6mmμ΄μƒμ˜ 골쑰직이 μ‘΄μž¬ν•˜μ˜€λ‹€. 3.μ •μ€‘κ΅¬κ°œλ΄‰ν•©λΆ€μœ„μ— κ΅μ •μš© μž„ν”Œλž€νŠΈλ₯Ό μ μš©ν•  λ•Œ μ•ˆμ •μ μΈ μ‹λ¦½λΆ€μœ„λŠ” κ΅¬κ°œν‰λ©΄μ„ λ”°λΌμ„œ ANSλ‘œλΆ€ν„° λ‚¨μžλŠ” 19.43mm ν›„λ°©λΆ€μœ„, μ—¬μžλŠ” 17.62mm ν›„λ°©λΆ€μœ„μ— ν•΄λ‹Ήν•˜μ˜€λ‹€. μ΄μƒμ˜ 결과둜 μ •μ€‘κ΅¬κ°œλ΄‰ν•©λΆ€μœ„μ˜ μ•ˆμ •μ μΈ μ‹λ¦½λΆ€μœ„μ—μ„œλŠ” κ΅μ •μš© μž„ν”Œλž€νŠΈ 식립이 μ ν•©ν•˜λ‹€κ³  νŒλ‹¨λ˜λ©° 이λ₯Ό λ°”νƒ•μœΌλ‘œ μ •μ€‘κ΅¬κ°œλ΄‰ν•©λΆ€μœ„μ˜ κ΅μ •μš© μž„ν”Œλž€νŠΈλ₯Ό ν™œμš©ν•˜μ—¬ κ΅μ •μΉ˜λ£Œμ˜μ—­μ„ λ„“νž 수 μžˆμ„ 것이닀. [영문]Since the miniscrew stability is mainly obtained from the cortical bone, it is important to take this into consideration to improve miniscrew stability. In this point, miniscrews implanted in the maxilla have less stability than that in the mandible due to the porous bony structure. However, the midpalatal suture area composed of dense cortical bone has been determined as the best anchorage site in the maxilla. But there is lack of data for the amount of vertical bone in the midpalatal region for miniscrew implantation. The purpose of this study was to measure the structure of the midpalatal suture area using CT image and V-works 4.0 program(Cybermed Inc., Seoul, Korea). CT images of 14 male and 14 female adults were reconstructed. In detail, it was 1) to measure the length of maxilla on the midsagittal plane 2)to measure vertical bone height in the midpalatal area 3) to establish the zone of safety for miniscrew implantation. The following results were obtained. 1. In the length of ANS-PNS, the mean length was 51.08mm in males and 47.34mm in females. There was a statistically significant difference between male and female. (p<0.05) 2. The vertical bone height of the midpalatal suture area was above 6mm except 15mm posterior from the central point of ANS-PNS in males. 3. The zone of safety was located 19.43mm posterior from the ANS in males while it was 17.62mm in females upon the palatal plane. These results support that the safety zone of the midpalatal area is suitable for screw implantation. Midpalatal miniscrew implantation is a powerful tool in modern orthodontics. Through many applications, it can expand the modern orthodontic field.ope

    Improvement in VAT with real taxation - Focusing on incorrect invoice -

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