Need to Take Into . . . Towards the Empirical . . . How Can We Explain . . . Scale-Invariance Ideas Scale-Invariance: . . . Which Dependencies . . . Explain the Empirical Need to Go Beyond . . . Which Value a Should . . . Soil-Water Characteristic What Is the Resulting . . . Curve Home Page Title Page Edgar Daniel Rodriguez Velasquez 1 , 2 and Vladik Kreinovich 3 ◭◭ ◮◮ 1 Universidad de Piura in Peru (UDEP), edgar.rodriguez@udep.pe ◭ ◮ Departments of 2 Civil Engineering and 3 Computer Science University of Texas at El Paso, El Paso, Texas 79968, USA, Page 1 of 37 edrodriguezvelasquez@miners.utep.edu, vladik@utep.edu Go Back Full Screen Close Quit
Need to Take Into . . . 1. Need to Take Into Account Water Content Towards the Empirical . . . How Can We Explain . . . • It is important to make sure that the roads retain suf- Scale-Invariance: . . . ficiently stiff under all possible weather conditions. Which Dependencies . . . • Out of different weather conditions, the most impor- Need to Go Beyond . . . tant effect on the road stiffness is produced by rain. Which Value a Should . . . What Is the Resulting . . . • Rainwater penetrates: Home Page – the reinforced-soil foundation of the pavement Title Page (called subgrade soil ) ◭◭ ◮◮ – that underlies more stiff layers of the road. ◭ ◮ • The presence of water decreases the road’s stiffness. Page 2 of 37 Go Back Full Screen Close Quit
Need to Take Into . . . 2. Towards the Empirical Formulas Towards the Empirical . . . How Can We Explain . . . • The mechanical effect of water can be described by the Scale-Invariance: . . . corresponding pressure h . Which Dependencies . . . • In transportation engineering, this pressure is known Need to Go Beyond . . . as suction . Which Value a Should . . . What Is the Resulting . . . • This pressure is easy to explain based on every person’s Home Page experience of walking on an unpaved road. Title Page • When the soil is dry, it exerts high pressure on our feet. ◭◭ ◮◮ • This prevents shoes from sinking, and keepd the sur- ◭ ◮ face of the road practically intact. Page 3 of 37 • On the other hand, when the soil is wet, the pressure drastically decreases. Go Back • As a result, the shoes sink into the road, and leave deep Full Screen tracks. Close Quit
Need to Take Into . . . 3. Towards the Empirical Formulas (cont-d) Towards the Empirical . . . How Can We Explain . . . • Similarly, the car’s wheels sink into a wet road and Scale-Invariance: . . . leave deep tracks. Which Dependencies . . . • The effect is not so prominent on paved roads, but still Need to Go Beyond . . . moisture affects the road quality. Which Value a Should . . . What Is the Resulting . . . • We want to describe this effect in quantitative terms. Home Page • This will enables us to predict the effect of different Title Page levels of water saturation. ◭◭ ◮◮ • For this, we need to find the relation between the water ◭ ◮ content and the suction. Page 4 of 37 • Usually, for historical reasons, this effect is described as the dependence of water content θ on suction h . Go Back • We can also invert this dependence and consider the Full Screen dependence of suction h on the water content θ . Close Quit
Need to Take Into . . . 4. Towards the Empirical Formulas (cont-d) Towards the Empirical . . . How Can We Explain . . . • The dependence of θ on h is known as the soil-water Scale-Invariance: . . . characteristic curve (SWCC, for short). Which Dependencies . . . • Until the 1990s, this dependence was described by a Need to Go Beyond . . . power law θ = c · h − m for some parameters c and m > 0. Which Value a Should . . . What Is the Resulting . . . • Since the suction decreases with the increase in water Home Page content, the exponent − m should be negative. Title Page • This law works reasonably well for intermediate values of θ . ◭◭ ◮◮ ◭ ◮ • However, this formula is not perfect. Page 5 of 37 • For example, for θ → 0, this formula implies the phys- ically unreasonable infinite value of suction pressure. Go Back • When θ is large, the soil is heavily saturated with wa- Full Screen ter. Close Quit
Need to Take Into . . . 5. Towards the Empirical Formulas (cont-d) Towards the Empirical . . . How Can We Explain . . . • In this case, the power law is also not in good accor- Scale-Invariance: . . . dance with the empirical data. Which Dependencies . . . • To get a better fit with the observations, researchers Need to Go Beyond . . . proposed a more complex formula Which Value a Should . . . � − c . What Is the Resulting . . . ln( e + ( h/a ) b ) � θ = const · Home Page • An even more accurate description comes if we multi- Title Page ply θ ( h ) by an additional factor: ◭◭ ◮◮ � 1 + h � ◭ ◮ ln h r C ( h ) = 1 − � . Page 6 of 37 � 1 + h 0 ln Go Back h r Full Screen Close Quit
Need to Take Into . . . 6. How Can We Explain the Empirical Formula? Towards the Empirical . . . How Can We Explain . . . • At present, there is no theoretical explanation for this Scale-Invariance: . . . formula. Which Dependencies . . . • Without a theoretical explanation, we cannot guaran- Need to Go Beyond . . . tee that this formula is indeed the best. Which Value a Should . . . What Is the Resulting . . . • In general, the presence of a theoretical explanation Home Page increases our confidence in an empirical formula. Title Page • From this viewpoint, it is desirable to come up with a theoretical explanation for this formula. ◭◭ ◮◮ ◭ ◮ • Such an explanation is provided in this talk. Page 7 of 37 Go Back Full Screen Close Quit
Need to Take Into . . . 7. Scale-Invariance: Reminder Towards the Empirical . . . How Can We Explain . . . • Let us start with a general problem of finding the de- Scale-Invariance: . . . pendence y = f ( x ) between physical quantities. Which Dependencies . . . • From the physical viewpoint, we need to take into ac- Need to Go Beyond . . . count that: Which Value a Should . . . – the same physical quantity What Is the Resulting . . . Home Page – can be represented by different numerical values. Title Page • A specific value depends on what measuring unit we select. ◭◭ ◮◮ • For example, we can measure distances in meters or ◭ ◮ kilometers. Page 8 of 37 • The same distance will be represented by different num- Go Back bers: 2 km becomes 2000 m. Full Screen • With pressure characteristics (like suction), we can use Close Pascals or we can the US unit psi (pounds/inch 2 ). Quit
Need to Take Into . . . 8. Scale-Invariance (cont-d) Towards the Empirical . . . How Can We Explain . . . • In general: Scale-Invariance: . . . – if we replace the original measuring unit with a Which Dependencies . . . different unit which is λ > 0 times smaller, Need to Go Beyond . . . – all numerical values get multiplied by λ : Which Value a Should . . . What Is the Resulting . . . x → x ′ = λ · x. Home Page • In many physical situations, there is no selected mea- Title Page suring unit. ◭◭ ◮◮ • So the formulas should not depend on what measuring ◭ ◮ unit we use. Page 9 of 37 • Of course, we cannot simply require that the formula Go Back remains exactly the same if we change the unit for x . Full Screen • That would means that f ( x ) = f ( λ · x ) for all λ and x Close – and thus, that f ( x ) = const: no dependence. Quit
Need to Take Into . . . 9. Scale-Invariance (cont-d) Towards the Empirical . . . How Can We Explain . . . • In reality, if we change the unit for x , we need to ap- Scale-Invariance: . . . propriately change the unit for y . Which Dependencies . . . • For example, the formula y = x 2 for the area y of a Need to Go Beyond . . . square with side x remains valid: Which Value a Should . . . What Is the Resulting . . . – if we switch from meters to centimeters, Home Page – but then we need to also change the measuring unit for area from square meters to square centimeters. Title Page ◭◭ ◮◮ • So, the desired property takes the following form: ◭ ◮ – for each λ > 0, there should exist a value µ > 0 such that Page 10 of 37 – if y = f ( x ), then y ′ = f ( x ′ ), where Go Back x ′ def = λ · x and y ′ def Full Screen = µ · y. Close • This property is known as scale-invariance. Quit
Need to Take Into . . . 10. Which Dependencies Are Scale-Invariant? Towards the Empirical . . . How Can We Explain . . . • Substituting y ′ = µ · y and x ′ = λ · x into the formula Scale-Invariance: . . . y ′ = f ( x ′ ), we get µ · y = f ( λ · x ). Which Dependencies . . . • Here, we have y = f ( x ), so f ( λ · x ) = µ · f ( x ). Need to Go Beyond . . . Which Value a Should . . . • Taking into account that µ depends on λ , we get the following expression: f ( λ · x ) = µ ( λ ) · f ( x ) . What Is the Resulting . . . Home Page • Small changes in x should cause equally small changes Title Page in y . ◭◭ ◮◮ • So the dependence f ( x ) must be smooth (differentiable). ◭ ◮ • From the above formula, we can conclude that the func- tion µ ( λ ): Page 11 of 37 Go Back – is equal to the ratio of two differentiable functions µ ( λ ) = f ( λ · x ) Full Screen and f ( x ) Close – is, thus, differentiable too. Quit
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