text: The property also is expected to depreciate by $85,000 this year. Like expense ratio, it is expressed in percentage. Service charges 4. Operating expenses can include: Operating expenses can also include the cost of goods sold, which are the expenses directly tied to the production of goods and services. The operating expense ratio (OER) is the cost to operate a piece of property compared to the income the property brings in. Because managing vacancies are included in efficient property management, including vacancies in an OER gives a more accurate picture of operating expenses and shows where improvements may be made. The offers that appear in this table are from partnerships from which Investopedia receives compensation. Funds from operations, or FFO, refers to the figure used by real estate investment trusts (REITs) to define the cash flow from their operations. Cost of goods sold (COGS) is defined as the direct costs attributable to the production of the goods sold in a company. If the business is scalable, the owner may increase the rent on each unit without greatly increasing operating expenses. Operating expense ratio is a financial tool business leaders use to evaluate the effectiveness in generating profit from operating expenses. The operating expense ratio (OER) is a profitability ratio. This includes all revenues from a piece of real estate. Let us take the example of a company named ADG Ltd which is engaged in the business of manufacturing electronic parts for Tier I auto parts supplier. Two companies can have the same operating ratio with vastly different debt levels, so it is important to compare debt ratios before coming to any conclusions. How to Calculate and Analyze a Company's Operating Costs, Apple Inc. Condensed Consolidated Statements of Operations (Unaudited), The operating ratio shows the efficiency of a company's management by comparing the total operating expense of a company to. The capitalization rate is the rate of return on a real estate investment property based on the income that the property is expected to generate. For example, a poorly managed property will most likely have higher vacancy rates, which will be reflected in the OER. Expense ratio (expense to sales ratio) is computed to show the relationship between an individual expense or group of expenses and sales. It is usually stated in terms of the percentage of the fund's assets that is taken out of the fund each year to cover its costs. An expense ratio of 1% per annum means that each year 1% of the fund's total assets will be used to cover expenses. It is a very popular ratio to use in real estate, such as with companies that rent out units. What Does the Operating Expense Ratio Measure? The cost incurred includes the raw material cost of$200 million, the direct labor cost of $120 million, the manufacturing overhead cost of$50 million, the selling expense of $30 million and the administrative expense of$10 mil… Operating expenses relate to the operations of a business entity. The operating expense ratio is simply the total of all expenses needed to operate a property divided by the potential rental income for a property. The total cost of sales (or cost of goods sold) was $37.00 billion while total operating expenses were$9.59 billion. The measure is very common in real estate analysis, whereby analysts are measuring the costs to operate a piece of property versus the income it … Apple. We also reference original research from other reputable publishers where appropriate. Along with return on assets and return on equity, it is often used to measure a company's operational efficiency. It is used to estimate the investor's potential return on investment in the real estate market. The operating ratio shows how efficient a company's management is at keeping costs low while generating revenue or sales. The operating expense ratio allows investors and analysts to understand how efficiently a business is able to produce goods or supply services. The cost of … One of our next finance KPI examples, the operating expense ratio (OER), shows the operational efficiency of your company by comparing operating expenses (the cost associated with running your core operations) to your total revenue. Operating expenses are costs associated with the operation and maintenance of the property. A lower operating expense ratio (OER) is more desirable for investors because it means that expenses are minimized relative to revenue. Therefore, the two costs must be added together to form the numerator in the operating ratio calculation. In other words, operating expenses include all administration and distribution expenses … Additional operating expenses that investors should figure into the OER include property management fees, landscaping, attorney fees, landlord’s insurance, and basic property insurance. What is the Operating Expense Ratio (OER)? As of 2017, it … These include white papers, government data, original reporting, and interviews with industry experts. Some companies take on a great deal of debt, meaning they are committed to paying large interest payments, which are not included in the operating expenses figure of the operating ratio. The operating ratio shows the efficiency of a company at keeping costs low while generating revenue. The measure is very common to approach in real estate analysis, whereby experts are estimating the expenses to work a bit of property versus the pay it produces. … The OER is used for comparing the expenses of similar properties. The calculation for the operating ratio is: OperatingRatio=OperatingExpenses+CostofGoodsSoldNetSalesOperating\, Ratio = \frac{Operating\, Expenses\, +\, Cost\, of\, Goods\, Sold}{Net\, Sales}OperatingRatio=NetSalesOperatingExpenses+CostofGoodsSold. Expense ratio is expressed in percentage. Here are the most common examples of revenue sources: 1. An investor should look for red flags, such as higher maintenance expenses, operating income, or utilities that may deter him from purchasing a specific property. It is computed by dividing a particular expense or group of expenses by net sales. OER is used for comparing the expenses of similar properties. How do you optimize your operating expenses? The net sales for Blue Trust Inc. are $5,000. This is a useful metric in measuring operational efficiency and can be leveraged in investigating any industry. JPMorgan Chase operating expenses for the quarter ending September 30, 2020 were$19.173B, a 22.62% decline year-over-year. The ratio is expressed in percentage. Operating Expense Ratio is the ratio between the cost of operation to the net revenue and is typically used in evaluating real estate properties, where higher Operating Expense ratio means higher operating expense as compared to its property income and serves as a deterrent and lower operating expense ratio implies lower operating costs and therefore, preferable and investment-friendly. In 2000, the asset-weighted average expense ratio for actively managed U.S. open-end mutual funds and ETFs was just above 1%. The cost of goods sold which are not included in the operating expenses is $1,000. The cap rate simply represents the yield of a property over a one-year time horizon (assuming the property is purchased on cash and not on loan). The operating expense ratio is determined by taking total expenses and dividing by total rental income. Understanding Operating Expense Ratio (OER), Formula for Operating Expense Ratio (OER), Operating Expense Ratio (OER) vs. There are two drawbacks to the OER for real estate investors. Investment analysts have many ways of analyzing company performance. Capitalization Rate, Limitations of the Operating Expense Ration, Understanding the Debt-Service Coverage Ratio (DSCR), EBITDA – Earnings Before Interest, Taxes, Depreciation, and Amortization. OER = Total operating expenses / Gross revenue Variable costs include daily operational expenses (labour, feed, crop protection, fuel, maintenance, insurance, repairs, and ag specialist fees, for example). An operating ratio that is decreasing is viewed as a positive sign, as it indicates that operating expenses are becoming an increasingly smaller percentage of net sales. Operating Expense Ratio The Operating Expense Ratio (OER) is the ratio of the total operating expenses and the effective gross income of the property. It is calculated by dividing a property's operating expense (minus depreciation) by its gross operating income. It is calculated by subtracting depreciation from a property’s operating expenses and then dividing by its gross revenue. The capitalization rate is used in the world of commercial real estate to indicate the rate of return that is expected to be generated on a real estate investment property. Find total operating expenses, which should be farther down the income statement. Return on sales (ROS) is a financial ratio used to evaluate a company's operational efficiency. The annual expense ratio is formulated by a simple accounting calculus: divide a fund's operational expenses by its average net assets. Operating expense ratio can be explained as a way of quantifying the cost of operating a piece of property compared to the income brought in by that property. The formula for calculating OER is simply operating expenses for a given period divided by gross profit for the same period. We calculate the numerator of the operating ratio by adding$37.00 billion (COS) + $9.59 billion (operating expenses) for a total of$37.9 billion for the period. When owning an apartment building, an investor should figure in vacancies by using effective rental income, or potential rental income minus vacancy and credit losses, rather than potential rental income. During 2018, the company clocked a total revenue of $450 million. This ratio gives you a direct comparison of your expenses to your income, allowing you to track your efficiency. EBITDA, or earnings before interest, taxes, depreciation, and amortization, is a measure of a company's overall financial performance. A high OER is unfavorable to profitability. The net operating income formula is calculated by subtracting operating expenses from total revenues of a property.As I mentioned earlier, revenues include more than just rental income. The operating ratio for Blue Trust Inc. is 80%. All of these line items are listed on the income statement. Expressed as a percentage, the net profit margin shows how much of each dollar collected by a company as revenue translates into profit. The offers that appear in this table are from partnerships from which Investopedia receives compensation. Companies must clearly state which expenses are operational and which are designated for other uses. Operating ratio (also known as operating cost ratio or operating expense ratio) is computed by dividing operating expenses of a particular period by net sales made during that period. These include all fees and costs incurred as the normal costs of doing business. The operating ratio is used for company analysis in various industries while the OER is used in the real estate industry. Operating expense ratio The operating expense ratio measures an operation’s variable costs relative to that operation’s gross revenue. In addition, the OER can show where potential issues may occur, such as utility bills increasing substantially, so investors can solve problems more quickly and protect their profit levels. Therefore, the annual OER can be calculated as: [($50,000×12)−85,000](65,000×12)=66%\frac{[(\$50,000 \times 12) - 85,000]} {(65,000 \times 12)} = 66\%(65,000×12)[($50,000×12)−85,000]=66%. Also, companies will typically not include non-operating expenses in the operating ratio. The relationship can be represented mathematically as follows: Operating Ratio = {Expense (or group of expenses) / Net Sales} * 100 However, most companies separate operating expenses from the cost of goods sold. When viewed over time, the operating expense ratio can also reveal if management is able to expand operations without dramatically increasing expenses. The expense can be an individual expense or a group of expenses like cost of goods sold, labor costs, material expenses, administrative expenses, or sales and distribution expenses. It's also important to compare the operating ratio with other firms in the same industry. Another ratio you can derive from operating costs is the operating expense ratio (OER). Thus, the OER should be used in conjunction with something like the capitalization rate when evaluating a property investment. Rental income 2. It is useful to track the operating ratio over a period of time to identify trends in operational efficiency or inefficiency. Second, because depreciation can be calculated in several different ways, the OER can be gamed by using a more favorable method of accounting for depreciation. In real estate, the operating expense ratio (OER) is a measurement of the cost to operate a piece of property, compared to the income brought in by the property. It is defined by the formula: Cap rate=net operating income÷current market value\text{Cap rate} = \text{net operating income} \div \text{current market value}Cap rate=net operating income÷current market value. "Apple Inc. Condensed Consolidated Statements of Operations (Unaudited)," Page 1. Formula: Operating ratio is computed as follows: The basic components of the formula are operating cost and net sales. In corporate finance, the debt-service coverage ratio (DSCR) is a measurement of the cash flow available to pay current debt obligations. If companies can make enough money from their operations to support the business, the company is usually considered more stable. This means that operating expenses consume approximately two-thirds of revenues generated by this property. An operating ratio that is going up is viewed as a negative sign, as this indicates that operating expenses are increasing relative to sales or revenue. Also, investors can monitor operating expenses and cost of goods sold (or cost of sales) separately to determine whether costs are either increasing or decreasing over time. An expense ratio is an annual fee charged to investors to cover the operating and administrative expenses of mutual funds and exchange-traded funds (ETFs). The investor also pays $50,000 for operating expenses including his monthly mortgage payments, taxes, utilities, and so on. Operating Ratio vs. Operating Expense Ratio. Calculate the operating ratio for the company. Here’s the operating expense ratio formula: The operating expense ratio formula measures how much of a property’s potential rental income is consumed by expenses needed to operate the property. Property management fees, utilities, trash removal, maintenance, insurance, repairs, property taxes, and other costs are included in OERs. As with any financial metric, the operating ratio should be monitored over multiple reporting periods to determine if a trend is present. You can learn more about the standards we follow in producing accurate, unbiased content in our. The smaller the ratio, the more efficient the company is at generating revenue vs. total expenses. As explained by Investopedia, the operating expense ratio (OER) is a helpful tool in carrying out the comparisons between the expenses of analogous properties. Investors must monitor costs to see if they're increasing or decreasing over time while also comparing those results to the performance of revenue and profit. Cost of goods sold can include the following: Revenue or net sales is the top line of the income statement and is the amount of money a company generates before expenses are taken out. If a company has a higher operating ratio than its peer average, it may indicate inefficiency and vice versa. For example, in the section above, we determined that the company’s operating income in the example was$452,000. Solution Use the below-given data for calculation of the operating ratio Therefore, the calculation of operating ratio is as follows, =(3000+1000)/5000 1. Operating ratio measures the relationship of expenses to sales. OER also does not take into account the market value of a property. Operating Expense Ratio (OER) Calculator. It is also known as an expenses-to-sales ratio. Apple's operating ratio must be examined over several quarters to get a sense of whether the company is managing its operating costs effectively. A company may need to implement cost controls for margin improvement if its operating ratio increases over time. Vending machines 5. The expense ratio of a stock or asset fund is the total percentage of fund assets used for administrative, management, advertising (12b-1), and all other expenses. A limitation of the operating ratio is that it doesn't include debt. Because it concentrates on core business activities, one of the most popular ways to analyze performance is by evaluating the operating ratio. Operating expense ratio. The operating expense ratio (OER) is defined as a measurement of the cost to operate a piece of property compared to the income brought in by the property. The operating ratio for Apple means that 63% of the company's net sales are operating expenses. JPMorgan Chase operating expenses for the twelve months ending September 30, 2020 were $102.366B, a 4.96% increase year-over-year. Conversely, if the operating ratio is falling, expenses are decreasing, or revenue is increasing, or some combination of both. It is equivalent to an organization’s operating expenses divided by its incomes. Overall, mid rise and high rise apartment buildings paid a greater share of their expenses for taxes, while garden apartments paid a higher share of expenses for salaries and personnel and repairs and maintenance. The operating expense ratio (OER) is calculated by dividing all operating expenses less depreciation by operating income. OER=Total operating expenses−depreciationGross revenueOER = \frac{\text{Total operating expenses} - \text{depreciation}}{\text{Gross revenue}}OER=Gross revenueTotal operating expenses−depreciation. Operating expenses include overhead expenses such as sales, general, and administrative costs. Take a hypothetical example, where Investor A owns a multi-family apartment building and brings in$65,000 per month in rent. It takes into account both the production and income side, as well as the expense side, to paint a clear picture of your operation’s efficiency. Accessed Oct. 19, 2020. The operating expenses are $3,000. For garden apartments, operating expenses were studied by the age of the buildings, with buildings 20+ years old reporting the highest operating expense increases, at 3.4%. An expense ratio measures the operational costs of a mutual fund relative to the fund's average net assets. Finally, as with all ratios, it should be used as part of a full ratio analysis, rather than in isolation. The operating expense ratio is a measure of your operation’s financial efficiency and shows how much you spend to generate income. An example of overhead might be the expense of the corporate office for a company because although necessary, it's not directly tied to production. The operating expense ratio (OER) is equal to a company's operating expenses divided by its revenues . Operating Expense Ratio Example Therefore, the investor may lose more money the longer they hold the property. If a property’s costs increase annually at a greater rate than income, the OER increases annually as well. The operating expense ratio (OER) is used in the real estate industry and is a measurement of what it costs to operate a property compared to the income that the property generates. The operating ratio is calculated as follows:$37.9 billion / $59.68 billion, which equals 0.63 or 63%. Parking fees 3. These costs help run the property on a daily basis. First, because it does not include the market value of a property, it does not inform an investor about the relative value of a property at purchase or sale. The operating expense ratio is used in real estate to measure the proportion of expenses to revenues for an income-producing property. In real estate, the operating expense ratio (OER) is a measurement of the cost to operate a piece of property, compared to the income brought in by the property. It is calculated by dividing a property's operating expense (minus depreciation) by its gross operating income. Learn more. The operating ratio shows the efficiency of a company's management by comparing the total operating expense (OPEX) of a company to net sales. Often referred to as the "cap rate," this measurement is computed based on the net income which the property is expected to generate. Definition and Explanation: Operating expenses ratio shows operating expenses as a percentage of sales. Operating costs are expenses associated with normal business operations on a day-to-day basis. Some companies list revenue as net sales because they have returns of merchandise from customers whereby they credit the client back, which is deducted from revenue. The gross income multiplier is obtained by dividing the property's sale price by its gross annual rental income, and is used in valuing commercial real estates, such as shopping centers and apartment complexes. Apple reported total revenue or net sales of$59.68 billion for the period. A low OER means less money from income is being spent on operating expenses. Benefits and wages for the production workers. OER finance calculator helps to find the operating cost of a product / service by net sales or income of the company. An income statement is one of the three major financial statements that reports a company's financial performance over a specific accounting period. Companies can sometimes cut costs in the short term, thus inflating their earnings temporarily. Operating-Expense ratio is a measurement of financial efficiency and is determined based on information derived from a business or farm operation’s financial statements specifically using the financials that determine gross farm income. Calculating OERs over a number of years may help an investor notice a property’s trends in operating expenses. The operating expense ratio (OER) is used in the real estate industry and is a measurement of what it costs to operate a property compared to the income that the property generates. operating expense ratio definition: the relationship between operating expenses and gross income . For this reason, loan payments, capital improvements, and personal property are excluded from operating expenses. From a company's income statement take the total. A property is more acceptable to an investor when the operating expense ratio is as low as possible. You will also need to calculate the property's depreciation expense, which will vary by the particular accounting method employed. The operating profit margin ratio is a key indicator for investors and creditors to see how businesses are supporting their operations. Please note that some companies include the cost of goods sold as part of operating expenses while other companies list the two costs separately. The ideal OER is between 60% and 80% (although the lower it is, the better). On the other hand, the operating ratio is the comparison of a company's total expenses compared to the revenue or net sales generated. While the cap rate is similar to OER in terms of measuring the profitability of an investment property, it differs from the OER in that it uses gross revenue rather than net income and places that in the denominator. A limitation of the operating ratio is that it doesn't include debt. Operating expenses are the costs associated with running the business that is not directly tied to the production of the product or service. Financial efficiency refers to how effectively a business or farm is able to generate income. The operating expense ratio is calculated by dividing total operating expenses by net sales. In order to calculate the OER for a property, you need to know the operating expenses. 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text:
# In Mathematica, is there a way to set a variable to the lowest value in a matrix?

It seems that a major aspect of this question is figuring out whether or not I can search the matrix and compare values.

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If $A$ is a matrix, Min[A] gives the smallest entry.

format: plain


text:
# Base Macro to Determine Control Type

Is there a property or method of a control that reliably reveals its type? For example, can I determine if a given object is a form, table control, list box or text box? I am trying to simplify some of my subroutines and functions so that inside them, they can determine the control type and take different actions accordingly. Or better yet, write a function that receives a control object, and returns a string telling the type of the control so I could reuse the function throughout my code. I'm particularly interested at the moment to determine if a given object is a table control, but the other types would be useful to determine as well.

I did finally figure out how to use MRI, so I can generally view the methods and properties of any object. I'm just not sure how to test for the type of a control. I suppose I could find some property that is unique to a table control or list box and check whether those methods/properties exist or something like that.

It would also be cool if there was a way to determine the format of the data in a control (kind of the same as the first question/paragraph). For example, is it numerical, currency, text, date, time, etc.? In some of my functions I pass a string parameter that tells the type in advance, like sFormat = "string", "currency", "long", "integer", "double", "float", "listbox", "null", etc. If there was a way to determine the data type programmatically, I could skip having to tell the functions what the type is, which would be nice.

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Reliably; Funny guy. Of course there is.

Each control model has a service name property. That is what you want

Here is an example routine which will walk down every control on a form looking for certain types of controls. And then go over what is happening. (actually it should be fairly self explanatory)

sub setNativeWidgetLook( aDataForm as variant, aYesNo as Boolean )
dim enControlModels
dim oControlModel as object

'thisComponent.LockControllers    '

if enControlModels.hasMoreElements then
do
oControlModel = enControlModels.nextElement
if oControlModel.ServiceName = "stardiv.one.form.component.Form" then
setNativeWidgetLook( oControlModel, aYesNo )
else
if oControlModel.ServiceName <> "stardiv.one.form.component.Grid" then
if oControlModel.ServiceName <> "stardiv.one.form.component.Hidden" then
oControlModel.NativeWidgetLook = aYesNo
end if
else
setNativeWidgetLook( oControlModel, aYesNo )
end if
end if
loop while enControlModels.hasMoreElements = True
end if

'thisComponent.UnLockControllers    '
end sub

Of course a dataform can nest include other dataforms (ie. for a master/slave type form) so in these lines:

if oControlModel.ServiceName = "stardiv.one.form.component.Form" then
setNativeWidgetLook( oControlModel, aYesNo )

if the current control is another dataform the routine uses recursion to walk down the controls on this sub-dataform first.

Then it looks it checks that the control model is neither a grid control nor a hidden control (which is not just any control hidden, it is separate type of control) and if not then it sets the property NativeWidetLook (yes for OS, no for LO).

If the current control was a grid, that again has a collection of controls so again it uses recursion to call itself and walk down those also.

You will want to get a copy of SDK from the LO download site, with that is the documentation that will give you the needed ServiceNames, etc and this document can be linked to from MRI so you can use it to get details while you are working.

When you say datatype of the control are you actually asking to find the dataytpe of the database field the control is linked to? (I'll assume you are)

so here is an example of another working routine which uses that:

sub PrettyControl( aControl as variant, aReqColor as double, setDisplaySize as boolean )
if  aControl.ServiceName = "stardiv.one.form.component.Edit" _
or  aControl.ServiceName = "stardiv.one.form.component.TimeField" _
or  aControl.ServiceName = "stardiv.one.form.component.DateField" _
then
if aControl.BoundField.isNullable = 0  and aReqColor <> -1 then
aControl.BackgroundColor = aReqColor
end if
if aControl.BoundField.TypeName = "VARCHAR"_
OR aControl.BoundField.TypeName = "CHAR"_
then
if setDisplaySize then
aControl.MaxTextLen = aControl.BoundField.DisplaySize
end if
end if
if aControl.HelpText = "" then
aControl.HelpText = aControl.BoundField.Helptext
end if
end if
end sub

Each data awara UI control has a property BoundField which is the database column and here the routine checks that for certain things like whether is can be null and if not colors the UI controls background, the only thing that might need explaining are the last lines:

if ...
more

Golden. While some of this is a bit above my head, I can perceive the answer is here if I slowly parse through. I like that there is a "service name property." All the stuff about recursion and whatnot is great additional info too. More than likely my situation will only require a subset of what you have offered as an answer, but for those who want to go deeper, you've provided them fodder as well. Thanks a lot!

( 2018-11-13 13:34:37 +0200 )edit

.hasMoreElements... nooice!

( 2018-11-13 13:40:57 +0200 )edit

My question concerned the Base control type. As in, when I "replace with" and choose text field, numerical, currency, etc. Obviously I don't have to use "replace with" to make new controls, but it's faster to CTRL+drag an existing control (to copy it), replace with to change its type, and change its properties. Then all the size, font and color stuff is consistent. Knowing the underlying type of the MySQL column is just as good or better, as long as I use the roughly matching control in Base.

( 2018-11-13 13:56:04 +0200 )edit

That help text thing is cool. Since I'm using MySQL, I wonder if it would grab the MySQL field descriptions. That would be kinda cool. Though most of my stuff doesn't require much user instruction. "This is an address field, so type an address in it." Heh heh.

( 2018-11-13 14:00:26 +0200 )edit

One of the reasons I want to know the type is so I can know which .updateXXX() method to use. For example .updateString(), .updateLong(). Same goes for .getXXX methods. Also, list boxes are unique & need .commit & not .boundField. I pass an array of fieldnames to functions & a matching array of field types. It's not terrible, and it works, but it would be cool if I didn't have to pass that extra array. I think I'll be able to write a function to determine the control types with your info. Sweet!

( 2018-11-13 14:09:08 +0200 )edit

Ah, well some real over kill on my part then - but knowing how to work with the boundField property at runtime can have its advantages.

( 2018-11-13 14:17:58 +0200 )edit

IIRC, you can not get to the MySQL table column descriptions at least not with this property. It would be nice if you could. Maybe there is a way, I'll ponder that a bit.

( 2018-11-13 14:20:48 +0200 )edit

One method to get MySQL field comments is through Information Schema:

SELECT column_name, column_comment
FROM INFORMATION_SCHEMA.COLUMNS
WHERE table_name='YOUR_TABLE_NAME'

This will get you all columns with their comments for a particular table.

( 2018-11-13 20:34:31 +0200 )edit

+1. Thanks! Also added a trailing ' (single quote) on comment lines to work around bug in Ask.Libreoffice.org which diesplay's comments wrong, so your code displays a bit better at this web site.

( 2018-11-14 21:12:29 +0200 )edit


text:
# Applying a cost function to a string based on a pattern string

I have a cost function Cost_Func(pt,STRING) that calculates value in a specific way based on the given pattern string pt and main string STRING.

A mathematical expression ( y := y-11*int(STRING[i-length]))/6+ int(STRING[i])*pow(11,length), this is the main calculation ) is used,

and the intuition is, it will be faster than KMP string matching (to compare I present here the function  KMPSearch(pt,STRING)),

but to my surprise, it is not.

I thought KMP function would have to use a lot of value-comparison, while the expression y runs through the loop once.

In the hope of better performance, I used assignment expressions in list comprehension (Python 3.8), but the function is still slower than the KMP matching function. I need to be faster than KMP function, otherwise the cost function will not be accepted.

txt = "010110101010101010100000000000111111111111111111111111111111111110000"
pat =  "01000000000001"
import time
def Cost_Func(pt,STRING):
x=y=0
h,length,cnst=[],len(pt),3
[(x := x+pow(11*int(pt[i]),(i+1)),y := y+pow(11*int(STRING[i]),(i+1))) for i in range(length)]
h=[((i-length+1)) for i in range(length,len(STRING)) if (y := (y-11*int(STRING[i-length]))/6+ int(STRING[i])*pow(11,length))>x/cnst]
return h
def KMPSearch(pt, tt):
k=[]
M = len(pat)
N = len(txt)
lps = [0]*M
j = 0
computeLPSArray(pat, M, lps)
i = 0
while i < N:
if pat[j] == txt[i]:
i += 1
j += 1
if j == M:
k.append(i-j)
j = lps[j-1]
elif i < N and pat[j] != txt[i]:
if j != 0:
j = lps[j-1]
else:
i += 1
return k
def computeLPSArray(pat, M, lps):
len = 0
lps[0]
i = 1
while i < M:
if pat[i]== pat[len]:
len += 1
lps[i] = len
i += 1
else:
if len != 0:
len = lps[len-1]
else:
lps[i] = 0
i += 1

k=200
t0 = time.time()
for i in range(1,k):
Cost_Func(pat,txt)
t1 = time.time()
for i in range(1,k):
KMPSearch(pat, txt)
t2 = time.time()
print("Cost_func:",(t1-t0)/k*10000,"KMP:",(t2-t1)/k*10000,k)

Is there a faster way to complete the Cost_Func(pt,STRING)?

For example, would library like NumPy help? Is there any other technique?

The source of the code of KMP function is taken from here.

• Is this function supposed to work only for strings made of 0s and 1s? Jun 30 at 19:40
• @IEatBagels yes, but the coefficient might vary, it is related to a Machine learning component, the idea is, everything is expressed in binary so, if we can calculate for binary string, we are actually doing it for all type of sting. Jun 30 at 20:11
• But, if I understood correctly, you'll have to convert your string to a binary string when you want to look for the pattern? If so, the time needed to do this would overshadow the performance gain you'd have from your algorithm, that's without considering the encoding issues when converting to binary string, no? Jun 30 at 20:14
• @Michael First: I implore you to find better sources of learning than Geeks for Geeks and "Md. Tahmid Hossain". They contain advice ranging from good, to bad, to aggressively bad. Jul 3 at 12:32
• Second: @200_success made a rollback with description After an answer has been posted, please do not alter the code or add rebuttals in the question. This was appropriate and your rollback of the rollback was not. Jul 3 at 12:33

Without saying this lightly, this code is a proper nightmare. Here is a list of things that do not make your code faster:

• Combining variable assignments on one line
• Writing a list comprehension used for its side-effects and then throwing away the result of the comprehension
• Removing all whitespace from comprehensions
• Hyper-abbreviating variable names

So don't do any of those. Add type hints, follow PEP8 for your variable and function names and whitespace, add a __main__ guard, add unit tests, expand your first comprehension to a plain loop, replace pow() with the ** operator, loop more pythonically via zip, make compute_lps_array() return instead of mutate, and you get what I consider to be a bare-minimum first refactor:

from typing import Literal, Sequence

PatternT = Sequence[Literal['0', '1']]

def cost_func(pattern: PatternT, string: PatternT) -> list[int]:
x = y = 0
length = len(pattern)
const = 3

for i, (p, s) in enumerate(zip(pattern, string), start=1):
x += (11 * int(p)) ** i
y += (11 * int(s)) ** i

h = [
(i - length + 1)
for i in range(length, len(string))
if (
y := (y - 11 * int(string[i - length])) / 6 + int(string[i]) * 11**length
) > x / const
]
return h

def kmp_search(pattern: PatternT, text: PatternT) -> list[int]:
i = j = 0
k = []
M = len(pattern)
N = len(text)
lps = compute_lps_array(pattern, M)

while i < N:
if pattern[j] == text[i]:
i += 1
j += 1
if j == M:
k.append(i - j)
j = lps[j - 1]
elif i < N and pattern[j] != text[i]:
if j != 0:
j = lps[j - 1]
else:
i += 1
return k

def compute_lps_array(pattern: PatternT, M: int) -> list[int]:
lps = [0] * M
len = 0
i = 1

while i < M:
if pattern[i] == pattern[len]:
len += 1
lps[i] = len
i += 1
elif len != 0:
len = lps[len - 1]
else:
lps[i] = 0
i += 1

return lps

def test() -> None:
text = "010110101010101010100000000000111111111111111111111111111111111110000"
pattern = "01000000000001"

assert cost_func(pattern, text) == [
1, 3, 5, 17, 18, 19,
20, 21, 22, 23, 24, 25, 26, 27, 28, 29,
30, 31, 32, 33, 34, 35, 36, 37, 38, 39,
40, 41, 42, 43, 44, 45, 46, 47, 48, 49,
50, 51
]
assert kmp_search(pattern, text) == [17]

if __name__ == '__main__':
test()

The KMP search doesn't lend itself well to Numpy vectorisation. If you really need this to be faster, don't use Python. If you want it to be Python-compatible, write it in C with an FFI to Python. It's probable that someone has already done this for you..

On the contrary, cost_func is definitely vectorisable. Recognise that your predicate y := ... > x/const is a recurrence relation, and one that has a pretty easy solution. In my very limited testing, this is equivalent:

def cost_func(pattern: np.ndarray, string: np.ndarray) -> np.ndarray:
x = (11. ** (np.nonzero(pattern)[0] + 1)).sum()
y0 = (11. ** (np.nonzero(string[:pattern.size])[0] + 1)).sum()

string[pattern.size:] * 11**pattern.size -
11/6 * string[:-pattern.size]
)

# Inspired by linear recurrence relation solution at
# https://stackoverflow.com/a/30069366/313768
pot = 6.**np.arange(string.size - pattern.size + 1)
revpot = pot[-2::-1]

const = 3
indices, = (ycomp > x/const).nonzero()
return indices + 1

def string_to_array(string: Sequence[Literal['0', '1']]) -> np.ndarray:
return np.array([
int(c) for c in string
])

def test() -> None:
pattern = string_to_array("01000000000001")
text = string_to_array("010110101010101010100000000000111111111111111111111111111111111110000")

assert np.all(
cost_func(pattern, text) == [
1, 3, 5, 17, 18, 19,
20, 21, 22, 23, 24, 25, 26, 27, 28, 29,
30, 31, 32, 33, 34, 35, 36, 37, 38, 39,
40, 41, 42, 43, 44, 45, 46, 47, 48, 49,
50, 51
]
)

if __name__ == '__main__':
test()

In the recurrence question I linked, there is another, simpler solution to express the recurrence relation with a single function call, scipy.signal.lfilter. I have not tested its performance, but I somewhat trust its numerical stability to be better and the results are the same:

def cost_func(pattern: np.ndarray, string: np.ndarray) -> np.ndarray:
x = (11. ** (np.nonzero(pattern)[0] + 1)).sum()
y0 = (11. ** (np.nonzero(string[:pattern.size])[0] + 1)).sum()
string[pattern.size:] * 11**pattern.size -
11/6 * string[:-pattern.size]
)

# Inspired by linear recurrence relation solution at
# https://stackoverflow.com/a/53705423/313768
ycomp, _ = scipy.signal.lfilter(
b=[1, 0], a=[1, -1/6], x=addends, zi=[y0/6],
)

const = 3
indices, = (ycomp > x/const).nonzero()
return indices + 1

Also on numerical stability: your quantities are enormous, even in this small example going past 4e14. Strongly consider operating in log-domain math instead.

• If one is getting " 'type' object is not subscriptable ": stackoverflow.com/a/63460173/3989930 Jun 30 at 17:09
• Speed isn't the point of the first refactor. It's disentangling the illegible code in the original post. Jun 30 at 18:20
• I am getting AttributeError: 'str' object has no attribute 'size' (imported numpy as np) so I replaced with sz_pt=len(pattern) and sz_tt=len(string) but I am getting string[sz_pt:] * 11**sz_pt - ....OverflowError: cannot fit 'int' into an index-sized integer, could you suggest how to solve it? Jun 30 at 21:16
• @Michael My mistake; I forgot to show invocation. Please see the edit. Jun 30 at 21:49
• Did you mean use of math.log(a,Base) when you said log-domain math? for example ycomp = math.log(ycomp,Base)? If I do that, will I be able to do Equality Comparison (==), for example, like ycomp == x/const ? Also, is there anything else we can do to improve (of course, your first code is faster than mine, I am exploring that, haven't seen 2nd one) besides manipulation related to vectorization? Jul 2 at 14:37


text:
# 529 Calculator

Created by Carolyn Kovacs and Oghenekaro Elem
Reviewed by Wojciech Sas, PhD and Jack Bowater
Last updated: Oct 26, 2022

This 529 calculator will help you determine the rising cost of education – from kindergarten to college – and the required monthly contribution you need to make to a 529 plan to afford the education cost in the coming years. If you have a child or ward, you are probably already thinking about how you will pay for their education 🎓. The information below will explain how you can achieve this with 529 savings plans. The great thing about 529 plans is that they have no age limits 🧓, so you can even set up a 529 plan for yourself if you need it! Use our 529 plan calculator to estimate the savings based on your contribution to the 529 savings plan when you begin investing in future education.

By taking advantage of a 529 plan, you will enjoy the benefits of growing your investment tax-free, and you may even eliminate the need for a student loan in the future. But just in case someone might need it, we have a student loan calculator to help you in situations. If you have already made it into college - congratulations! You may want to check out our college GPA calculator💯 to keep track of your grades while you study. If you keep reading, you will learn answers to common questions ❓ like:

• What is a 529 account?
• How does a 529 plan work?
• What are qualified education expenses you can sponsor with a 529 plan?
• How to calculate college savings?

## The 529 account - What is a 529 plan?

A 529 plan is a savings plan that allows you to save and grow your investment towards paying for education tax-free📈.

The name "529 plan" comes from the fact that these plans are authorized by Section 529 of the Internal Revenue Code created in 1996. Legally known as a , these plans allow you to contribute to a 529 savings account where interest is tax-deferred. That means, unlike regular savings or investment accounts where you are required to pay taxes on interest returns to the IRS annually, nobody taxes your 529 accounts! 💰 And when you eventually want to make withdrawals or take distributions from the 529 accounts for qualified education expenses, you don't pay any federal taxes.

But, you have to understand all the 529 rules to take full advantage of a 529 plan. For instance, you need to pay attention to for the tax advantage.

## Types of 529 savings plan

There are two types of 529 plans:

• Education savings plans; and
• Prepaid tuition plans.

The majority of 529 plans are education savings plans, which involves investing into a 529 college savings account which will be withdrawn to pay for educational costs. Education savings plans are primarily used in saving for college; however, additions to the plan now allow it to pay for K-12 tuition 👶, computers 💻, student loan payments, and even some culinary and cosmetology schools. You can find a list of all eligible schools on the .

Prepaid tuition plans allow you to prepay all or part of a participating college or university tuition costs at the current price for future enrollment. That means you can pay for the future cost of education at today's price for the beneficiary of the 529 savings accounts. These plans are usually state-sponsored for residents in the state and do not include K-12 education. The Private College 529 Plan is also a separate prepaid plan for private colleges sponsored by private colleges.

## What are qualified education expenses?

The government first designed the 529 education plan to cater for "qualified higher education expenses", and so the plan targeted college savings. Then, in 2019, it was expanded from "qualified higher education expenses" to just "qualified education expenses" so that anyone can save for:

1. Pre-college K-12 education tuition. You can now save up to $10,000 annually for school tuition from kindergarten to 12th grade 🚸. 2. The total cost of apprenticeship programs and trade schools ⚒. 3. Student loan repayments, limited to$10,000. You can only pay off $10,000 with the 529 plan if a student loan is more than$10,000 📉.

4. The total cost of college tuition 👩🎓👨🎓, enrolment, attendance fees, and associated costs of:

• Mandatory textbooks and supplies 📚;
• A computer, internet access, and software needed for course work;
• Room and board expenses, such as rent and housing costs within the limits of the "cost of attendance" figures reported by the college; and
• Special needs equipment for students with disabilities.

As long as you use the distributions from the 529 accounts for the expenses above, they are qualified to be withdrawn tax-free. But, if you use the distribution for an unqualified expense, the earning of that distribution will be taxed according to your income tax bracket, and you will pay a 10% penalty.

## How does a 529 plan work?

When you open a 529 plan, the state sponsoring the 529 plan determines the plan's rules and limits and chooses a plan manager to invest the money in a portfolio of mutual funds, stocks, or other securities that can generate returns. Depending on your risk appetite and investment goal, you can choose an investment portfolio from options provided by the plan manager. State agencies or educational institutions can also sponsor the 529 plan.

Like most investment accounts, you have to pay fees on the 529 fund management. It's a good thing to look out for a portfolio's performance and expenses when choosing a 529 plan. Unlike most investment accounts, like the 401(k) and IRA, where you have to pay taxes when withdrawing, you may have to pay taxes on the money you contribute to the 529 accounts, but don't have to pay taxes when making a withdrawal. In this sense, a 529 savings plan is similar to a Roth IRA, which helps you save money tax-free for your retirement.
And in case you'd like to know more, we suggest you check out our roth ira calculator.

Most people who understand its tax advantage use it as a college savings plan; that's why you may also refer to our 529 calculator as the 529 college savings plan calculator. Therefore, you can use the 529 savings calculator to determine how much money you would save over time by putting your college savings account into a 529 college savings plan.

Since different states have their own rules and limits for the 529 education plan they sponsor, there can be some differences in each state's 529 plan. Gladly, you don't necessarily have to invest in the state where you live or where you want your child to go to school, so it is good to which offers the most tax benefit, lower fees or better investment returns.

## How to use the 529 calculator and select the best 529 plan?

Our 529 savings calculator will let you determine the cost and required contribution to save for education cost for as long as 12 years! (K-12 👶😉)

1. First, you need to provide current age and age at start of school of the 529 account beneficiary.

2. Then, select the number of school years the 529 plan will pay for. For instance, if you're saving for a four-year college program, you can select 4 years. Or you can select up to 12 years if you want to start early, saving for your child's K-12 years.

3. The calculator will compute the number of years to first school year and the required years to make deposits into the 529 plan. You can change the values to see how different scenarios work in the 529 calculator's advanced mode.

4. Next, fill in the school program's current annual cost and cost inflation. This will help assess how the cost increases over the years.

5. You may already have an initial investment for the 529 plan. Input the figures. If you don't, no worries, it won't stop your progress.

6. Provide your assumed annual investment return to evaluate how the 529 fund portfolio could perform over the years. It is essential to remember that you won't always get the same investment return every year. Investment returns fluctuate up and down depending on the performance of the portfolio you've chosen, but you can be confident that using this tool will keep you on track if you use the expected average.

7. Finally, suppose you want to compare how the 529 plan performs vs. regular investment or savings account, i.e., one subject to income tax. In that case, you can select the marginal tax bracket you belong to in advanced mode to see how it impacts the results on the savings chart 📊. The education cost annual breakdown table 📑 gives an overview of each year's cost and the required contribution needed to save for that particular year, leading up to the year that you eventually stop contributing to the 529 plan.

If you have multiple 529 plans, you can quickly compare how they will perform in different situations and choose the best 529 plan that aligns with your goal.

If you're not sure about your marginal tax bracket, check out the to ascertain what marginal rate applies to your income level. In actuality, taxes in a savings account are calculated cumulatively with your total income for a year, which may differ from the estimated value provided in the savings chart 📊. For instance, if you happen to be in a 12% tax bracket, you will pay 10% on the first $9,950 of your annual income + 12% on the remaining income you've earned that is less than$40,525. You can learn more about how taxation works on the IRS website linked above.

## How to calculate college savings? 529 fund example

Assuming you have a 5-year-old child and you plan to start saving for a college education. You intend to make an initial investment of $1,000 and set aside$3,000 every year in the future for a four-year degree program when they come of age. How do you know if the money you will be saving from your monthly salary will be sufficient to meet the education cost considering inflation and the rising cost of education by that time?

Let's say you want to invest the funds in a portfolio that returns 8% interest per annum. You would need to determine the number of deposits you will have to make leading up to the child's first year of college at the age of 18 and the cost of college in their final college year to ascertain if you need to make extra contributions on your $250 (which is $3000/12) per month contribution. You can carry out the calculation in the following steps:

Step 1: Determine the number of investment years you will be saving for

Current age = 5 years

Age to start college = 18 years

Years to first college year (freshman) = Age to start college - Current age = 18 – 5 = 13 years

Number of college years = 4

Years to make deposits in the 529 plan = Years to start first school year + Number of school years = 13 + 4 = 17 years

Next, you will need to know the current annual cost of a college program and its cost inflation rate.

Current annual cost of college at a preferred college equals $12,000. Note: the current annual cost (in this case,$12,000) is what you would pay today for a college year. But since prices tend to increase with inflation, you need to account for the possibility that college costs might go up.

Cost inflation = 4% = 0.04

Initial investment = $1000 Expected annual investment return = 8% = 0.08 Step 2: Calculate the total college cost for the four years the child will be in college Adapting the future value of annuities formula to calculate the total college cost: $\text {Future Value} = \text {(Present Value} \times (( 1 + \text I )^\text {number of periods} - 1 ) / \text I) - \text{(Present Value} \times (( 1 + \text I )^{number of periods}- 1 ) / \text I)$ Thus, $\text {Total college cost} = \text {total college cost until the last contribution year} - \text {total cost until the first college year}$ $\begin{split} & \text {Total college cost} = \text{(current cost} \\ & \times (1 + \text {inflation})^{\text{number of years to make deposit}} - 1) \\ & / \text {inflation} – (\text{current cost} \times \\ & ((1 + \text {inflation})^{\text{number of years to start first school year}} - 1) / \text {inflation}) \end{split}$ $\text{Total college cost} = (12,000 \times ((1 + 0.04)^{17} - 1) / 0.04) – (12,000 \times ((1 + 0.04)^{13} - 1) / 0.04)$ $\text{Total college cost} = \284,370.15 \ – \ \199,522.05 = \84,848.10$ Step 3: Calculate the cost for each college year to determine the contribution needed per year Adapting the future value (FV) formula on the current cost of college: $\text{Future Value} = \text{Present Value} \times ( 1 + I )^{\text{number of periods}}$ Thus, $\small \begin{split} & \text{Cost year n} = \text {current cost} \\ & \times (1 + \text{inflation})^{\text{Years until n}^\text{th} \text { college year}} \end{split}$ $\text{Cost year one} = 12,000 \times (1 + 0.04)^{13} = \19,980.88$ For the subsequent years: $\text{Cost year two} = 12,000 \times (1 + 0.04)^{14} = \20,780.12$ $\text{Cost year three} = 12,000 \times (1 + 0.04)^{15} = \21,611.32$ $\text{Cost year four} = 12,000 \times (1 + 0.04)^{16} = \22,475.77$ Step 4: Find out how much is required to save for each year per month For the n-th year; $\small \begin{split} & \text {Required contribution year n} = \\ & (\text{Cost year n} - (\text {initial investment} \\ & \times (1 + \text{interest per month})^{\text{total number of months}})) \\ & / (( (1 + \text{interest per month})^{\text{Years to n}^{\text{th}} \text{ year} + 1} - 1 ) \\ & / \text{interest per month}) \end{split}$ $\text{Interest per month} = \text{annual interest per annum} / 12$ $\text{Interest per month} = 8\% / 12 = 0.67\% = 0.0067$ $\text{Total number of months} = \text{Years to make deposit} \times 12$ $\text{Total number of months} = 17 \times 12 = 204\text{ months}$ Therefore, $\begin{split} \text {Required contribution year one} &= \\ \frac {(19,980.88 - (1,000 \times (1 + 0.0067)^{204}))} {(( (1 + 0.0067)^{13 + 1} - 1 ) / 0.0067)} \end{split}$ For the subsequent years: $\begin{split} \text {Required contribution year two} &= \\ \frac {(20780.12 - (1,000 \times (1 + 0.0067)^{204}))} {(( (1 + 0.0067)^{13 + 2} - 1 ) / 0.0067)} \end{split}$ $\begin{split} \text {Required contribution year three} &= \\ \frac {(21,611.32 - (1,000 \times (1 + 0.0067)^{204}))} {(( (1 + 0.0067)^{13 + 3} - 1 ) / 0.0067)} \end{split}$ $\begin{split} \text {Required contribution year four} &= \\ \frac {(22,475.77 - (1,000 \times (1 + 0.0067)^{204}))} {(( (1 + 0.0067)^{13 + 4} - 1 ) / 0.0067)} \end{split}$ $\begin{split} & \text {Total required contribution} =\\ & \52.28 + \48.84 + \45.80 + \43.07 \\ & = \189.98 \text{ monthly} \end{split}$ Since you only need to contribute $189.98 per month, it means that the $3,000 you intend to set aside annually will be more than sufficient to meet the college cost, offering an extra$60.02 per month to the required deposits.

You can avoid making a huge payment or taking a loan that will accrue huge compounding interest to pay for your child's education in the future by using the 529 plan. But, carrying out all the steps to arrive at the exact figure you need to save can be tiresome, which is why the 529 college saving calculator is a fantastic tool to help you figure things out in seconds. Interestingly, unlike other college fund calculators that determine how to calculate college savings without accounting for withdrawals from the savings account, this 529 calculator accounts for the fact that you will be taking out annual distributions to pay for education costs each year.

If you put the same amount of money into a standard investment or regular savings account with the same return on investment, you would have to pay taxes on any money you make on the investment. Assuming that you belong to the lowest tax bracket and only pay a 10% income tax, the amount you would accumulate in a taxable account in the same number of years is around $79,555, which is about 7% less than the calculated savings goal. And the higher your tax bracket, the more you pay in taxes, the less money you will have in a taxable account after paying your taxes. ## FAQ ### What is a 529 account? A 529 account is a savings account under the authorized 529 plan that allows you to save and grow your investment towards paying for education tax-free. ### Are 529 contributions tax-deductible? No. Contributions to investment accounts are tax-deductible if you have to pay tax on them at withdrawal, but you fund the 529 accounts with after-tax money, so you don't pay tax when withdrawing from the account. However, some states may offer the benefit to deduct your 529 contributions from your taxable income. ### Is room and board qualified education expense? Yes. Room and board expenses within the limits of the "cost of attendance" figures reported by a 529 participating college are qualified education expenses. ### How to open a 529 plan? • Research the . • Select the best 529 plan which offers the most benefits in returns, less fund management fees, and penalty if you ever decide to opt-out. • Enroll in a 529 plan and complete the . Make sure you ask questions to clarify anything you're unsure of. • Fund the account and choose your preferred investment option. • Monitor and adjust the performance of your 529 accounts. ## How much money should I invest in a 529 plan? In 2020, the average 529 plan balance was$25,664, which was an all-time high, and the trend shows growth almost every year for the past decade. However, only with a freshman entering college in 2017 used a 529 college savings plan to help pay for their child's education. Only about one-third of Americans even know that a 529 plan is an available educational savings tool. So, the first step is educating yourself on what a 529 plan is (which you are doing right now - so great job!) and checking out our 529 plan calculator to get an idea of what you could save. Keep in mind that you can use the 529 calculator as a 529 college fund calculator as well.

The amount you invest is up to you as most states have minimum initial investment requirements. However, because growth is exponential, the more you can set away early on (and be consistent with annual payments!), the higher your overall return on your investment will be. Play around with our 529 calculator to see how an early start can affect your overall savings. It is also interesting to compare this with our student loan payment calculator, which shows how much extra you may have to pay if you have to take out a loan with interest to pay for college.

## Is there any reason I shouldn't start a 529 college savings plan?

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# How to prove scaled delta function relation mathematically?

I am working through Shankar's Introduction to Quantum Mechanics. I have come across exercise 1.10.1, which asks the reader to show that:

$$\delta(ax)=\frac{\delta(x)}{|a|}.$$

I can understand it intuitively, in the sense that $$x\to ax$$ will 'compress' the delta function in the x-axis direction, and it has to be $$|a|$$ because the delta function is an even function.

However, how can this relationship be shown mathematically? I have been playing with it for a few days and I seem to be struggling.

Edit:

To expand on the method I have been trying, I am starting with the following completeness relation that Shankar gives:

$$\int_{-\infty}^{\infty}\vert x'\rangle\langle x'\vert dx'=I$$

I then create a new variable, $$x''$$, such that:

$$x''=ax'$$ $$dx''=adx'$$

then I substitute that in:

$$\int_{-\infty}^{\infty}\vert x''\rangle\langle x''\vert \frac{dx''}{a}=I$$

then I perform his operation (that he uses to show $$\int\delta(x)=1$$), where he takes the inner product on one side with $$\vert f \rangle$$ and on the other with $$\langle x \vert$$:

$$\int_{-\infty}^{\infty}\langle x \vert x''\rangle\langle x''\vert f \rangle\frac{dx''}{a}=\langle x \vert I \vert f \rangle = f(x)$$

$$f(x)\int_{-\infty}^{\infty}\delta(x,x'') dx'' = a f(x)$$

However, this seems to lead to the opposite conclusion, that $$\delta(ax)=a\delta(x)$$.

Where have I gone wrong?

## migrated from physics.stackexchange.comSep 20 at 18:24

This question came from our site for active researchers, academics and students of physics.

• Put it in an integral, make a variable substitution, consider cases for when $a>0$ and $a<0$ – Aaron Stevens Sep 19 at 18:46
• Would Mathematics be a better home for this question? – Qmechanic Sep 19 at 18:48
• @AaronStevens I have been trying to do that, but I don't quite see how. – Time4Tea Sep 19 at 19:01
• I thought $$\int_{-\infty}^{\infty}\langle x \vert x''\rangle\langle x''\vert f \rangle\frac{dx''}{a}=f(x)\int_{-\infty}^{\infty}\delta(x,x'') \frac{dx''}{a}$$ – Aaron Stevens Sep 19 at 19:21
• @AaronStevens that seems equivalent to what I got. However, doesn't that mean that $\delta(ax)$ is a factor of $a$ bigger than $\delta(x)$, because you have to divide it by $a$ to get the same result? Perhaps I'm interpreting it wrong ... – Time4Tea Sep 19 at 19:24

You're working much too hard. The delta function is defined by

$$\int_a^b \delta(x) f(x) dx = \begin{cases}f(0) & 0 \in (a,b) \\ 0 & \text{otherwise}\end{cases}$$

Start with $$\int_a^b \delta(ax)f(x) dx$$, perform a variable substitution with $$u= ax$$, and then apply the definition of the delta function. As Aaron said, you should separately consider the cases when $$a<0$$ and $$a>0$$, and you should arrive at the result right away.

• Thanks a lot, this is really helpful. This problem has been bugging me for days ... – Time4Tea Sep 19 at 21:27

J. Murray's answer was very helpful. So, just to make sure I've got this right, this is the definition of the delta function:

$$\int_a^b \delta(x) f(x) dx = \begin{cases}f(0) & 0 \in (a,b) \\ 0 & \text{otherwise}\end{cases}$$

and we want:

$$\int_b^c \delta(ax) f(x) dx$$

So, substitute $$u=ax$$, $$du=adx$$:

$$\int_{ab}^{ac} \delta(u) f(u/a) \frac{du}{a}=?$$

$$\frac{1}{a}\int_{ab}^{ac} \delta(u) f(u/a) du=\begin{cases}\frac{f(0)}{a} & 0 \in (ab,ac) \\ 0 & \text{otherwise}\end{cases}$$

by the definition of the delta function. Then, we have to switch the variables back:

$$\frac{1}{a}\int_{b}^{c} \delta(ax) f(x) a dx=\begin{cases}\frac{f(0)}{a} & 0 \in (b,c) \\ 0 & \text{otherwise}\end{cases}$$

$$\int_{b}^{c} \delta(ax) f(x) dx=\begin{cases}\frac{f(0)}{a} & 0 \in (b,c) \\ 0 & \text{otherwise}\end{cases}$$

which is what we want; however, that assumes that $$a>0$$. If $$a<0$$:

$$\frac{1}{a}\int_{ab}^{ac} \delta(u) f(u/a) du=\begin{cases}-\frac{f(0)}{a} & 0 \in (ac,ab) \\ 0 & \text{otherwise}\end{cases}$$

because $$ab$$ is now larger than $$ac$$, so the integral is 'reversed'. So, that is what gives the general case of:

$$\int_{b}^{c} \delta(ax) f(x) dx=\begin{cases}\frac{f(0)}{|a|} & 0 \in (b,c) \\ 0 & \text{otherwise}\end{cases}$$

Is that correct?

• Looks right to me! – J. Murray Sep 19 at 21:32


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## How To Work Out Angle In Trigonomotry

Angles of Elevation and Depression, Applications, Trigonometry, Maths, Year 9, NSW Suppose that you are looking at an object in the distance. If the object is above you, then the angle of elevation is the angle your eyes look up. If the object is below you, the angle of depression is the angle …

Angles of Elevation and Depression, Applications, Trigonometry, Maths, Year 9, NSW Suppose that you are looking at an object in the distance. If the object is above you, then the angle of elevation is the angle your eyes look up. If the object is below you, the angle of depression is the angle …

Now that we know what an angle is, let's think about how we can measure them. And we already hinted at one way to think about the measure of angle in the last video where we said, look, this angle XYZ seems more open than angle BAC.

Three-dimensional trigonometry problems can be very hard and complex, Well, look at the square base diagram. If we can work out the side length of the square base, we can use Pythagoras’ Theorem to work out the length of AF. So let’s work on the side length of the base. The only unused piece of information in the question is the 72° angle. Let’s have a look at that triangle: Its

How to work out angle in trigonomotry

To find the angle you use the arctangent function like this, angle $=\tan^{-1}\left(\frac{y}{x}\right)$. But I am curious, is this the only way to know the angle... But I am curious, is this the only way to know the angle...

Angles of Elevation and Depression, Applications, Trigonometry, Maths, Year 9, NSW Suppose that you are looking at an object in the distance. If the object is above you, then the angle of elevation is the angle your eyes look up. If the object is below you, the angle of depression is the angle …

Measure the three sides for each of two right triangles that form the compound angle. For the triangle with angle A, use the ruler or tape measure to measure adjacent side a, opposite side a …

Example 3 Applying geometry and trigonometry with angle of elevation From a point A ,aman observes that the angle of elevation of the summit of a hill is 10 .He then walks towards the …

Now that we know what an angle is, let's think about how we can measure them. And we already hinted at one way to think about the measure of angle in the last video where we said, look, this angle XYZ seems more open than angle BAC.

19/10/2016 · The unknown angle, for this problem, is the acute angle opposite the 3 inch leg. Set up a trigonometry equation. In this case, because you know all three sides of the triangle, you actually have a choice of functions.

To find the angle you use the arctangent function like this, angle $=\tan^{-1}\left(\frac{y}{x}\right)$. But I am curious, is this the only way to know the angle... But I am curious, is this the only way to know the angle...

Angles of Elevation and Depression, Applications, Trigonometry, Maths, Year 9, NSW Suppose that you are looking at an object in the distance. If the object is above you, then the angle of elevation is the angle your eyes look up. If the object is below you, the angle of depression is the angle …

To find the angle you use the arctangent function like this, angle $=\tan^{-1}\left(\frac{y}{x}\right)$. But I am curious, is this the only way to know the angle... But I am curious, is this the only way to know the angle...

### How to work out angle in trigonomotry -

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# 16.8: When the Solute Obeys Henry's Law, the Solvent Obeys Raoult's Law

In Section 16.4, we conclude that any sufficiently dilute solute obeys Henry’s law, and define a hypothetical, pure-liquid standard state that makes the solute activity equal to its mole fraction, $$\tilde{a}_A\left(P,y_A,y_B\right)=y_A$$. In Section 16.7, we find that the mole fractions and activities of the components of any binary solution are related by

$y_Ad \ln \tilde{a}_A + y_B d \ln \tilde{a}_B =0.$

For a solute that obeys Henry’s law, we have

\begin{align*} d{ \ln \tilde{a}_B\ } &=-\left(\frac{y_A}{y_B}\right)d{ \ln y_A\ } \\[4pt] &=-\left(\frac{y_A}{y_B}\right)\left(d{ \ln y_B\ }\right)\left(\frac{d \ln y_A}{d \ln y_B}\right) \\[4pt] &=-\left(\frac{y_A}{y_B}\right)\left(d \ln y_B\right)\left(\frac{dy_A/y_A}{dy_B/y_B}\right) \\[4pt] &= -\left(\frac{y_A}{y_B}\right)\left(d \ln y_B\right)\left(\frac{-dy_B/y_A}{dy_B/y_B}\right) \\[4pt] &= d \ln y_B \end{align*}

This result follows for any choice of standard state for the activity of solvent $$B$$. It is satisfied by $$\tilde{a}_B=ky_B$$, where $$k$$ is a constant. It is valid even if $$A$$ is completely nonvolatile. When gas-phase $$B$$ behaves as an ideal gas, and we choose the ideal gas at $$P^o$$ as the standard state for both gas- and solution-phase B, we have

$\tilde{a}_B\left(\mathrm{gas}\right)=f_B={P_B}/{P^o}={x_BP}/{P^o}$

Since the standard states are the same, the fugacity and activity of $$B$$ in solution are the same as they are in the gas phase above it. We have $$\tilde{a}_B\left(\text{solution}\right)=ky_B={x_BP}/{P^o}$$. To find $$k$$, we consider the system comprised of pure $$B$$, for which $$y_B=x_B=1$$ and $$P=P^{\textrm{⦁}}_B$$. Substituting, we find $$k={P^{\textrm{⦁}}_B}/{P^o}$$. With this value for $$k$$,

$\tilde{a}_A\left(\mathrm{solution}\right)={P^{\textrm{⦁}}_By_B}/{P^o}={x_BP}/{P^o}$

so that

$y_BP^{\textrm{⦁}}_B=x_BP.$

This is Raoult’s law.

Thus when the solute obeys Henry’s law and the solvent behaves as an ideal gas in the gas phase above its solution, the solvent obeys Raoult’s law.

Evidently the converse is also true. If the solvent obeys Raoult’s law, $$y_BP^{\textrm{⦁}}_B=x_BP$$. With pure ideal gas $$B$$ as the standard state for $$B$$ in both the gas phase and the solution phase, we have

$\tilde{a}_B\left(\text{solution}\right)=\tilde{a}_B\left(\mathrm{gas}\right)=f_B={P_B}/{P^o}={x_BP}/{P^o}=y_B\left({P^{\textrm{⦁}}_B}/{P^o}\right)$ so that $d{ \ln \tilde{a}_B\left(\mathrm{solution}\right)\ }=d{ \ln y_B\ }$

From $$y_Ad{ \ln \tilde{a}_A\ }+y_Bd{ \ln \tilde{a}_B\ }=0$$ and $$d{ \ln \tilde{a}_B\ }=d{ \ln y_B\ }$$, we have

$d{ \ln \tilde{a}_A\ }=-\left(\frac{y_B}{y_A}\right)d{ \ln y_B\ }=-\left(\frac{y_B}{y_A}\right)\left(\frac{dy_B}{y_B}\right)=\frac{dy_A}{y_A}=d{ \ln y_A\ }$

so that $$\tilde{a}_A\left(\mathrm{solution}\right)=ky_A$$, where $$k$$ is a constant. When we choose the standard state such that $$\tilde{a}_A\left(\mathrm{ss,\ }\mathrm{solution}\right)=1$$ when $$y_A=1$$, we find $$k=1$$ and $$\tilde{a}_A\left(\mathrm{solution}\right)=y_A$$. The activity of the solute is related to its fugacity and the fugacity of its standard state by

$\tilde{a}_A\left(\mathrm{solution}\right)=y_A=\frac{f_A\left(\mathrm{solution}\right)}{f_A\left(ss,\ \mathrm{solution}\right)}$

When $$y_A=1$$, the fugacity is that of the standard state, which is a system of the hypothetical pure liquid in equilibrium with its own ideal gas. Letting the pressure of this ideal gas be $${\textrm{ĸ}}_A$$, we have $$f_A\left(ss,\ \mathrm{solution}\right)={\textrm{ĸ}}_A$$, so that $$f_A\left(\mathrm{solution}\right)={\textrm{ĸ}}_Ay_A$$, which is equal to the fugacity of the gas with which it is at equilibrium. The fugacity of the ideal gas is $$x_AP$$, so that

$x_AP={\textrm{ĸ}}_Ay_A.$

This is Henry’s law. Thus, if solvent $$B$$ obeys Raoult’s law, solute $$A$$ obeys Henry’s law.

This page titled 16.8: When the Solute Obeys Henry's Law, the Solvent Obeys Raoult's Law is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Paul Ellgen via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

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# American Institute of Mathematical Sciences

October  2018, 23(8): 2989-3021. doi: 10.3934/dcdsb.2017183

## Stability of dislocation networks of low angle grain boundaries using a continuum energy formulation

 1 Department of Mathematics, Hong Kong University of Science and Technology Clear Water Bay, Kowloon, Hong Kong 2 Department of Mathematics, University of Connecticut 341 Mansfield Road, Storrs, CT 06269, USA

Received  June 2016 Revised  June 2017 Published  July 2017

Fund Project: The first author is partially supported by the Hong Kong Research Grants Council General Research Fund 606313. The second author thanks HKUST for hospitality

Low angle grain boundaries can be modeled as arrays of line defects (dislocations) in crystalline materials. The classical continuum models for energetics and dynamics of curved grain boundaries are mainly based on those with equilibrium dislocation structures without the long-range elastic interaction, leading to a capillary force proportional to the local curvature of the grain boundary. The new continuum model recently derived by Zhu and Xiang (J. Mech. Phys. Solids, 69,175-194,2014) incorporates both the long-range dislocation interaction energy and the local dislocation line energy, and enables the study of low angle grain boundaries with non-equilibrium dislocation structures that involves the long-range elastic interaction. Using this new energy formulation, we show that the orthogonal network of two arrays of screw dislocations on a planar twist low angle grain boundary is always stable subject to both perturbations of the constituent dislocations within the grain boundary and the perturbations of the grain boundary itself.

Citation: Yang Xiang, Xiaodong Yan. Stability of dislocation networks of low angle grain boundaries using a continuum energy formulation. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 2989-3021. doi: 10.3934/dcdsb.2017183
##### References:

show all references

##### References:
(a) Schematic illustration of a twist boundary. (b) Dislocation network consisting of two orthogonally intersecting screw dislocation arrays on a low angle twist boundary. For this twist boundary, across each horizontal dislocation (with Burgers vector $\mathbf b_1$) in the upper grain, the atoms above it shift to the left by $b_1/2$ and those below it shift to the right by $b_1/2$ with respect to the lower grain; across each vertical dislocation (with Burgers vector $\mathbf b_2$) in the upper grain, the atoms on its left shift downward by $b_2/2$ and those on its right shift upward by $b_2/2$ with respect to the lower grain. The net effect is making the upper grain rotate in the counterclockwise direction with respect to the lower grain
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